HoM Toolbox, or Historiography and Methodology for Mathematicians

Amy Ackerberg-Hastings (MAA Convergence)


Have you been assigned to teach the history of mathematics and are wondering how to make sense of the subject matter? Do you want to better understand the nuances raised in history of mathematics sessions at academic conferences or in articles in academic journals and monographs? Are you interested in the history of mathematics and want to try researching and writing it yourself, perhaps even for publication? Might you be an experienced instructor who wants your history of mathematics students to think more intentionally about the subject or to tackle more in-depth research projects? Have you taken a history of mathematics course as an undergraduate and want to know more about what it would be like to pursue a career in the field?

The author researching and writing the history of mathematics.
Figure 1. Historians of mathematics often describe libraries and archives as
one of their “happy places.” In 2002, the author spent three months as a
Dibner Library Resident Scholar at the Smithsonian’s National Museum of American History.

This series is designed for readers in all of these situations and more. The initial installment offers a sort of crash course in how historians collect, analyze, and synthesize sources into a written account (historiography) and the techniques, principles, and approaches they use while writing history (methodology). In addition to outlining several of the useful “tools” available to historians, the first article includes a sample assignment that helps students articulate their own understandings of what it means to research and write the history of mathematics, originally developed by the author to prepare undergraduate history majors for embarking on their capstone papers that would demonstrate their ability to interpret past events through the analysis of primary-source evidence.

Other articles in the series introduce various approaches to historical interpretation, ways of making sense through evidence-based arguments that highlight different perspectives on the past and reveal the complexity of history. These pieces may be read in entirety, or readers may choose to focus on an approach they have encountered in scholarly articles or books or that sounds intriguing. The series thus provides the history of mathematics with a very brief text on historiography as well as a course reader in methodology, akin to works widely employed in history programs, such as Green and Troup [2016] and Tosh [2018]. The history of mathematics is not only about what happened in the past; rather, it also deals with why and how those events unfolded. Learn to engage in these conversations among historians by reading further.

Cover of The Houses of History, a reader on theoetical approaches edited by Anna Green and Kathleen Troup.Cover of Historians on History, a reader on theoetical approaches edited by John Tosh.
Figure 2. Two examples of readers on theoretical approaches to historical interpretation,
available from Manchester University Press and Routledge, respectively.

Is there an approach to historical interpretation you would like to learn more about? Are you a historian of mathematics who would like to share your approach with undergraduate and secondary students and instructors? Please contact the editors.


HoM Toolbox, or Historiography and Methodology for Mathematicians: Introduction

Amy Ackerberg-Hastings (MAA Convergence)


This installment of the HoM Toolbox provides an overview of the chief considerations and practices involved in researching and writing the history of mathematics. Contrary to often-used phrases such as “History tells us” this or ”History shows” that, the meaning of past events in not self-evident. Rather, historians gather evidence left behind by people of the past—including books, letters, written records, and photographs, but also physical objects and the built environment—analyze that evidence by dissecting individual sources and by comparing sources against each other, and draw conclusions about how events unfolded and why individuals or groups were significant. Frequently, the evidence can support multiple conclusions, so historians discuss and debate their interpretations with each other.

History is a professional academic discipline, and historians hold each other to high standards for ethics and competence, such as the guidelines established by the American Historical Association [2019]. Although historians are expected to use sound methods and theories as they attempt to understand the past, those methods and theories are not closely-guarded secrets. Rather, they are available for anyone to learn and use; while some people will have more aptitude for history than others—as is the case with any human endeavor—a chief prerequisite for becoming a historian of mathematics is the willingness to work hard. If you have read this far, you can develop the skills of a historian!

Participants in a 2011 joint conference of BSHM and CSHPM.
Figure 3. Professional historians of mathematics hold academic conferences to exchange information and ideas.
This joint meeting of the British Society for the History of Mathematics and the Canadian Society for History and Philosophy of Mathematics took place at Trinity College Dublin in 2011.
Photo by Tony Mann, distributed via the CSHPM/SCHPM Bulletin.

The rest of this introductory installment outlines some of the major arenas in which the hard work of learning to be a historian occurs. First, we define “history” and explore why readers of this series might want to gain proficiency with researching and writing the history of mathematics. Second, we ask how we know about the past by considering various categories of primary sources as well as techniques for drawing information from those sources. Third, we look at how historians put together all of the information they have gathered to make an argument about why the person, place, event, idea, or time period they are describing was historically significant. We will see that this process of interpretation is both creative and fully grounded in primary-source evidence. Fourth, we will briefly note that the field of history of mathematics has its own history and show how that history connects to historiography, the practices historians use to make sense of the past. Finally, we will provide an exercise in writing your personal philosophy of the history of mathematics that can be used in a classroom or for self-study. References to other discussions of why history of mathematics is a worthwhile enterprise will appear throughout the article. To dig more deeply into the nuts and bolts of historical research and writing, readers are referred to two outstanding online handbooks aimed at history students in general: [Cronon 2008–2009] and [Rael 2004].

HoM Toolbox, or Historiography and Methodology for Mathematicians: Introduction – What is History?

Amy Ackerberg-Hastings (MAA Convergence)


Just like mathematicians, historians strive to articulate precise definitions. Thus, words that may be used interchangeably in everyday life need to be clarified and distinguished. In particular, “history” and ”the past” are not the same thing. For historians, the past consists of everything that ever happened. It is humanly impossible to record and report every detail about any or all past events—imagine for a moment that you decide to write a history of your life by noting each thing you did and every word you said. But we all experience dozens of events in just a single day that are lost to our memories almost as soon as they happen . . . and even if you could keep track of every minute happening, your readers likely would be bored to death by your account, and documenting your life would take so long that you would no longer have time to actually live your life. The past, therefore, is not history.

In addition to being necessarily selective, history involves answering questions—most notably, the 5 Ws and 1 H: Who? What? When? Where? Why? How? The first four questions are mainly informational, so it may seem fairly straightforward to collect facts from the sources that were created during the time period a historian is studying. (We will see on the next two pages that this assumption is actually an oversimplification.) Indeed, the “how?” and “why?” questions are the ones that move us from the past to history, because people in the past rarely were self-aware enough to write down their true motives or to document all of the processes that influenced their actions. In other words, historians must make inferences from the sources in order to draw conclusions about the meaning and significance of past events. These conclusions, or interpretations, comprise history.

Infographic with the 5W1H question words used by historians of mathematics.
Figure 4. Historians of mathematics ask many, many questions.
Resource created for Teachers Pay Teachers by Mr. Dignan’s Desk.

The history of mathematics, then, is the effort to understand how and why mathematics developed over time through identifying a representative selection of primary-source evidence created in the past and relevant to the topic whose history is being written, putting the information documented by the evidence into context, identifying relationships between the evidence, and drawing conclusions about how and why mathematics unfolded as it did. These interpretations give us insights into how human beings think, act, and relate to each other. Doing history thus helps us discern not only how mathematics evolved into the forms we know today, but also why mathematics became the way it is—because human beings made choices about what to study, where to share their knowledge, who could be involved, how to connect abstractions with the physical world, why the subject matters, and so on. While historians of mathematics, like all historians, must rely on the memories of people from the past—memories that could be incomplete, faulty, or biased—they use critical reading and thinking skills to sort through and interpret the evidence. This is sober, thoughtful work grounded in the effort to understand; historians avoid making assumptions as much as possible.

Ivor Grattan-Guinness explaining the history of mathematics.One of Grattan-Guinness's famous articles on the difference between history and heritage in mathematics.
Figure 5. In obituaries such as the one posted by Bocconi University in Milan, Ivor Grattan-Guinness (1941–2014) was shown explaining the history of mathematics. He wrote numerous articles on the differences between “history of mathematics” and “heritage of mathematics,” including [Grattan-Guinness 2004].

Assumptions about the past are often connected to a shared sense of memory that is usually called “tradition” or “heritage”. Ivor Grattan-Guinness wrote several essays (such as [2004]) explaining the difference between history and heritage while challenging readers to create sound history; see also [Tosh 2021]. Heritage typically celebrates the perceived heroes and highlights of a culture's or nation's past, which has the positive effect of providing a community with a common identity. For instance, Americans like to remember that the entire United States was outraged by Japan's attack on Pearl Harbor and went on to fight a good and honorable war, in which any non-heroic or bad actions that happened before or during the conflict were forgotten (e.g., the antisemitism of Charles Lindbergh and Henry Ford). Selectively mining the past for heritage thus leads researchers into at least three pitfalls:

  1. Thinking of the past as we want it to have been (nostalgia) instead of asking what really happened, according to primary source evidence;
  2. Using assumptions about the past to guide behavior in the present (upholding supposed traditions);
  3. Critiquing past people for not thinking the way we do in the present (both a neglect of context and a belief in constant and inevitable progress).

In contrast, approaching the past with the mindset of a historian includes:

  1. Recognizing that what motivated people in the past may be different from what motivates people in the present;
  2. Considering people in their original political, social, economic, cultural, intellectual, and religious settings or context;
  3. Tracing patterns of change and continuity over time.

These activities begin during the collection and analysis of historical evidence, as we will see in the next section.

HoM Toolbox, or Historiography and Methodology for Mathematicians: Introduction – How Do We Know About the Past?

Amy Ackerberg-Hastings (MAA Convergence)


Sources of information about the past fall within three categories. Historians chiefly utilize primary sources, written, audiovisual, and other records created at or near the time of the events under study by the participants in the events. Several examples are shown below; note that primary sources can be created from a variety of materials—they are not all pieces of paper, and physical artifacts may be as informative as the words in written documents.

As historians process and interpret the evidence from primary sources, they create secondary sources, interpretations or reconstructions of the past. If they cannot access all of the primary sources relevant to their topic, historians may consult other secondary sources that discuss those materials. As a matter of courtesy and as the main means of furthering the conversations among historians that drive the discipline of history forward, authors will also acknowledge, extend, and critique the interpretations made by other historians in other secondary sources. We will revisit these conversations when we discuss approaches to historical interpretation. Another essential characteristic of a secondary source is that it will contain a thorough bibliographic trail (through footnotes and a bibliography) that allows other scholars to verify and evaluate the author's conclusions—so, historians “show their work” just like math students, and they invite their readers to join the conversations among historians.

Researchers may also come across tertiary sources, which are accounts of past events based entirely upon secondary sources without consultation of primary sources. These authors are simply sharing information that is already known without developing original interpretations of their own. Textbooks and encyclopedias are standard examples of tertiary sources, although popular history, documentaries, and historical fiction may also fall within this category.

The book, Newton's 1687 Principia Mathematica, is a primary source.The academic journal, American Journal of Mathematics, is a primary source.This letter by Florence Nightingale is a primary source.This manuscript with diagrams by a medieval student is a primary source.
This map drawn by Oronce Fine is a primary source.This newspaper clipping from the 18th century is a primary source.This geometric model from the late 19th century is a primary source.This photograph of the mathematician Li Shanlan with students is a primary source.
Figure 6. Some of the many types of primary sources in the history of mathematics,
as collected in Convergence’s Mathematical Treasures.

Just as you may have learned to do in a library skills class, historians assess the quality and reliability of all three categories of sources. What are the physical characteristics of a source, and what do those characteristics tell us about how and when it was made? How do we decide whether a source is authentic? (Feel free to take a break to check out this forgery in Mathematical Treasures.) What can we ascertain from the content of a source: are authors factual, free from error, explicit about their motives and biases, and so on? How does the source compare to other sources on the same topic?

History educators have developed a plethora of tools for collecting information from primary sources; two of my favorites are the (US-history-focused) websites History Matters by George Mason University and Getting Started with Primary Sources by the Library of Congress. I modified a worksheet from [Drake and Drake Brown 2003] for distribution to my own students as this Document Analysis Form and Object Analysis Form. (See [Ackerberg-Hastings 2019] for further discussion of primary-source analysis in the mathematics classroom.) The Convergence article, Online Museum Collections in the Mathematics Classroom [2014], offers a number of suggestions for teaching with historical mathematical objects, while [Wardhaugh 2010] guides readers through the unique challenges posed by making sense of historical mathematical writings.

An excellent case study from the history of mathematics for the analysis of primary sources—as well as the accidental way many primary sources survive to be discovered by researchers—is the Archimedes Palimpsest. It was copied in the 9th century, erased in the 13th century by thrifty scribes, re-discovered by Johan Ludwig Heiberg in the early 20th century, vanished (again) around the outbreak of World War I, returned to the public eye by an unidentified seller and buyer in 1998, and read through a variety of X-ray and digital techniques by a team assembled by the Walters Art Museum in Baltimore. The primary sources you work with may not have such a dramatic backstory, but they will still play an important role in helping you and your readers better understand the history of mathematics. On the next page, we will talk about taking the insights you gained from a set of primary sources and shaping them into a historical interpretation.

A portion of the Archimedes Palimpsest before imaging to uncover the scientific text.The same portion of the Archimedes Palimpsest after imaging, revealing a mathematical diagram.
Figure 7. Imaging techniques helped reveal the mathematics text and diagrams hidden beneath religious texts.
An introduction to the Archimedes Palimpsest is available in Convergence’s Mathematical Treasures.

HoM Toolbox, or Historiography and Methodology for Mathematicians: Introduction – How Do We Create History?

Amy Ackerberg-Hastings (MAA Convergence)


You have chosen a topic in the history of mathematics, perhaps read some tertiary sources to get a sense of the background and scope of what you want to find out about that topic, identified relevant primary and secondary sources, and gathered information and ideas from those sources—carefully noting the citation for each tidbit, of course, because you want to be just as reliable and professional as the authors of your secondary sources. Whew! This is where my students would suggest it was time for pizza and a tasty beverage, and they would groan audibly when I would grin and point out the hard work was just beginning. (My courses were in fact all online, but it is impressive how these reactions can nonetheless be transmitted electronically.) It is tempting to remain in the research stage indefinitely, but at some point every historian has to start writing. That is when efforts to interpret the evidence begin in earnest, as historians return to the W and H questions that led to their research projects.

Addressing those pesky “how?” and “why?” questions about the past usually requires historians to contemplate two fundamental patterns in human activity:

  • change and continuity over time;
  • the causes, courses, and consequences of historical events.

The concepts of causation and change help us understand what happened and explain why those happenings were historically significant—in other words, why we should remember those events and note their influence on subsequent incidents. Historians summarize their understandings and explanations in a thesis statement; they then expand their understandings and explanations into a fully-developed argument grounded in historical evidence and sound reasoning. I have already mentioned several excellent guides to constructing a historical argument [e.g., Cronon 2008–2009 and Rael 2004; alternative options include Schrag 2007–2022, whose “Organization” section is particularly strong; Brown n.d.; Conolly-Smith 2007; Storey 2020; and Booth, Colomb, and Williams 2016]. Here, we will focus on how the decisions historians make about causation and change inform their theoretical approaches to analyzing and interpreting the past.

Chart about Thinking Like a Historian, developed by Nikki Mandell and Bobbi Malone.
Figure 8. A chart depicting major components of historical thinking, developed on behalf of the Wisconsin Historical Society for elementary and secondary students and teachers by Nikki Mandell and Bobbie Malone [2007].

In history, “theoretical approach” is a term that denotes a conceptual lens or framework through which a historian views, understands, and draws conclusions about historical evidence—and, in turn, about causation and change. Since we are all shaped by our own experiences, education, and personal preferences, different historians may zero in on different aspects of the past and argue for the central importance of those aspects in creating causation and change. Those different perspectives denote different theoretical approaches. In other words, theoretical approaches are how historians get to their interpretations, not the end result of their research and not a guess about what happened in the past. Although there feasibly could be as many different theoretical approaches as there are historians, in practice historians tend to group by similar approaches into “schools” of historical interpretation. That simply means those historians hold similar ideas about what history is and how it ought to be researched. Some schools you may have heard about include Empirical approaches to historical interpretation, Marxist approaches, the Annales School, or Postmodernism. If you have attended history of math talks at an academic conference, you may have heard speakers criticizing “Whiggish” approaches to historical interpretation. If you would like to learn more about how theoretical approaches are central to the professional practice of history and find brief discussions of some of the major schools, see Gabrielle Spiegel’s 2008 presidential address to the American Historical Association [2009]. (More in-depth studies are provided by the readers mentioned earlier, [Green and Troup 2016; Tosh 2018].) In the next section, we will trace the development of some of the theoretical approaches that have been influential in the history of mathematics.

HoM Toolbox, or Historiography and Methodology for Mathematicians: Introduction – What is the History of the History of Mathematics?

Amy Ackerberg-Hastings (MAA Convergence)


Historiography is about how historians create history both now and throughout the passage of time. In other words, the professional discipline of history of mathematics has a history of its own. Historians of mathematics are human beings, influenced by their contexts and their own personalities, so neither the purpose nor the practice of history of mathematics has remained static. On this page, we briefly trace some of the major developments in the history of the history of mathematics, emphasizing changes in approaches to historical interpretation. While the local needs of their history of mathematics courses may reasonably dissuade instructors from lecturing on this material, they are welcome to direct students who ask questions about historiography to this page. As is true of mathematics, an exposure to the history of the history of mathematics can also humanize the subject by showing students that it has been researched and written by people who may not have been that different from them. (And, conversely, a look at “typical” historians of mathematics from the past can serve as a reminder that diversifying the discipline in the present remains a work in progress.) Anyone interested in reading further can find another concise overview in [Grattan-Guinness 1994]; two detailed academic studies are [Dauben and Scriba 2002] and [Remmert, Schneider, and Kragh Sørensen 2016].

Mathematicians have been acknowledging their predecessors since the ancient era, so historians often identify Eudemus of Rhodes (ca 350–ca 290 BCE) as the first historian of mathematics, although his writings have been lost and we only know about them because Pappus of Alexandria (ca 290–ca 350 CE) mentioned them. Mathematicians continued to discuss the work of earlier mathematicians who influenced their discoveries, but that historical information generally took the form of chronicles, lists of facts and events that lacked the analysis and interpretation that would have turned these accounts into histories. The historical information might also have been cherry-picked, naming only a few famous figures, even when mathematicians focused on a specific subject, such as John Wallis’s Treatise of Algebra.

John Wallis wrote about the history of mathematics in the 17th century.Jean Montucla wrote about the history of mathematics in the 18th century.Michel Chasles wrote about the history of mathematics in the early 19th century.
Figure 9. A few of the historians of mathematics included in Convergence’s Portrait Gallery:
John Wallis (1616–1703), Jean Étienne Montucla (1725–1799), and Michel Chasles (1793–1880).

Thus, Jean Étienne Montucla (1725–1799) is considered one of the first historians of mathematics to treat the subject as a scholarly enterprise and the first to prepare a history that covered all of mathematics. Like other Enlightenment writers, Montucla aimed for encyclopedic coverage and critically analyzed his subjects. His assessments were limited by a belief in progress that was also typical of the 18th century—for example, John Playfair’s shorter history of mathematics, prepared for Encyclopaedia Britannica, similarly assumed that mathematics was continually moving forward. Both men and others added regions such as India and China to the topics covered by history of mathematics, while professors such as Robert Simson tried to restore original sources from ancient Greece.

By the middle of the 19th century, the notion of periodization was well-established, as was the assumption that the history of mathematics was about its “great men”. Some scholars began editing the published works and unpublished manuscripts of giants such as Galileo or Euler. Although others looked at perceived differences between nations in order to express pride in or criticism for a country’s mathematical activities, histories persisted as largely descriptive endeavors, ignoring the complexities of the past. Near the end of the century, historians of mathematics increasingly adopted the first theoretical approach to historical interpretation that emerged from the newly-professionalized academic discipline of history, which was often characterized as “empirical” or “scientific.” For historians of mathematics, initially this meant that they paid very close attention to detail, producing massive volumes on narrow subjects, and that they studied mathematical works in their original languages.

Their interest in detail evolved in the histories of science and mathematics in the first half of the 20th century into an approach known as “internalism”. The term denoted a highly technical study of the mathematics in a document, with minimal consideration of the creators of mathematics, their institutions, and their careers, let alone their lives outside of mathematics and their social and cultural contexts. History of mathematics continued to develop in other ways, as its practitioners separated from the discipline of history of science, added areas of study such as mathematics education and medieval Arabic mathematics, and, after World War II, founded national and international academic societies and journals. Some historians began systematically documenting the participation of women in the history of mathematics. Courses in the history of mathematics became more common, which meant that textbooks had to be written for undergraduate students.

Florian Cajori wrote history of mathematics in the late 19th and early 20th centuries.Otto Neugebauer wrote history of mathematics in the 20th century.Bartel van der Waerden wrote history of mathematics in the 20th century.
Figure 10. More historians of mathematics from Convergence’s Portrait Gallery: Florian Cajori (1859–1930),
Otto Neugebauer (1899–1990), and Bartel Leendert van der Waerden (1903–1996).

From the perspective of theoretical approaches, major debates in the 1970s shaped how historians understood the purpose of history of mathematics and what they did to research and interpret the past. First, some historians of mathematics noticed that scholars in other fields were considering the contextual aspects that were ignored in internalism. They therefore advocated for an “externalism”; bitter battles between the two schools of interpretation were waged in journals such as Historia Mathematica, although ultimately internalists began taking into account more external factors while externalists became more mindful of internal characteristics. An early example of the new scholarship was [Mehrtens, Box, and Schneider 1981]. Adjunct to the fight between internalism and externalism, a historian of mathematics named Sabetai Unguru (b. 1931) took on leading mathematicians such as André Weil (1906–1998) and leading historians such as Bartel van der Waerden (1903–1996) over who was qualified to research and write the history of mathematics and whether it was appropriate to use modern notation and techniques to describe and analyze mathematical writings created before those ideas existed.

Beginning in the 1990s, historians of mathematics actively sought out “new” theoretical approaches from history and from history of science that addressed social, cultural, and institutional dimensions of the past. Some of these will be explored in later installments of this article series. Additionally, [Stedall 2012] evaluates some strengths and weaknesses of recent approaches and methodologies. A history of mathematics textbook that explains how historians’ understandings of the past have changed over time as new sources have come to light or as historians have looked at existing sources in new ways is [Barrow-Green, Gray, and Wilson 2019–2022].

Asger Aaboe wrote history of mathematics in the 20th century.D. T. Whiteside wrote history of mathematics in the 20th century.A mirror for self-reflection on researching and writing the history of mathematics.

Figure 11. Still more historians of mathematics from Convergence’s Portrait Gallery: Asger Aaboe (1922–2007) and Derek Thomas Whiteside (1932–2008). Can you envision yourself as a historian of mathematics, too?

HoM Toolbox, or Historiography and Methodology for Mathematicians: Introduction – Why Articulate a Philosophy of the History of Mathematics?

Amy Ackerberg-Hastings (MAA Convergence)


As students or instructors gain experience with researching and writing the history of mathematics, it can be useful to take some time to reflect on the issues raised on the previous pages of this article and articulate how they see themselves as historians of mathematics. I was instructed to complete such an exercise near the end of my first semester in a graduate history of technology and science program, and I subsequently used such an assignment in the courses I taught on historical methods and historical writing.

The general guidelines for the assignment (excluding specific formatting requirements) are reprinted below, but it is not necessary to replicate its depth or time commitment to reap the benefits of the concepts discussed in this article. Instructors of history of mathematics courses that include readings by professional historians may want to use the components of the assignment as student guidelines for evaluating the arguments in the readings. In courses with a research project, instructors could ask students to read this article and discuss the assignment questions as preparation for undertaking their projects. Undergraduates or faculty contemplating taking up the history of mathematics as a career path may better understand the discipline and themselves by preparing an essay independently or to share with a mentor. Because the objective is for individuals to define and elucidate their own philosophy of historical practice, there are many acceptable ways to approach and structure responses to the assignment.

Sample Assignment: Essay-Length Philosophy of the History of Mathematics

The Big Idea: Demonstrate how your understanding of historical practice and its theoretical foundations has evolved throughout this course by presenting a fully-developed philosophy of history of mathematics.

Format: Prepare a formal essay that sets out the conceptual framework you expect to employ as a participant in the professional academic discipline of history of mathematics, particularly while analyzing and synthesizing primary sources. In the course of your essay, comment on:

  • The distinction between "history" and "the past."
  • Why you study the history of mathematics (i.e., your conception of the purpose of history).
  • The role and reliability of evidence in history.
  • Why and how change occurs in history.
  • The relationship between objectivity and bias in historical interpretation.

Your response should be 1350–1650 words in length, plus footnotes and bibliography. It should be written in essay form, with an introduction paragraph containing your thesis statement, a body of approximately 5–6 paragraphs, a conclusion paragraph, footnotes, and bibliography.  Make your essay as mechanically perfect as you can manage: excellent spelling and grammar, no run-on sentences, no subject/verb disagreements, correct use of punctuation. Use active voice and past tense verbs. Cite any and all sources that you use.

Relevant Learning Outcomes: Explain what history is and why it matters that humans record it; Identify historical paradigms and distinguish how they affect interpretation and practice; Explain the strengths and limitations of various types, formats, and conditions of historical evidence; Employ the moral and ethical standards of the historical profession; Cultivate an identity as a historian.

Sample Scoring Rubric (based on 100 points):




Analysis (15%)

14–15 (Excellent)

Author directly addresses the questions of the assignment. Essay contains a clear argument-—i.e., lets the reader know exactly what the author is trying to communicate.

12–13 (Good)

Author competently addresses the questions of the assignment. An argument is present, but the reader must reconstruct it from the text.

10–11 (Needs Improvement)

Author attempts to address the questions of the assignment. Author attempts, but fails, to make an argument (e.g., starts with a rhetorical question or anecdote that is never put into context).

0–9 (Unacceptable)

Essay does not address the questions of the assignment. No attempt is made to articulate an argument.


Content/Evidence (20%)

18–20 (Excellent)

Provides compelling and accurate supporting evidence that convinces reader to accept main argument. The importance/relevance of all pieces of evidence is clearly stated. There are no gaps in reasoning—i.e., the reader does not need to assume anything or do additional research to accept main argument.

16–17 (Good)

Provides necessary supporting evidence to convince reader of most aspects of the main argument but not all. The importance/ relevance of some evidence presented may not be totally clear. Reader must make a few mental leaps or do some additional research to fully accept all aspects of main argument.

13–15 (Needs Improvement)

Insufficient evidence is provided to support author’s argument, or evidence is incomplete, factually incorrect, or oversimplified.

0–12 (Unacceptable)

Either no evidence is provided, or there are numerous factual mistakes, omissions or oversimplifications.


Professional Development & Reflection (15%)

14–15 (Excellent)

The essay includes original, thoughtful reasons why the author wants to study the history of mathematics. These reasons also indicate the author's sophistication in thought and expression has deepened throughout the semester. The author presents a professional demeanor and shows interest and enthusiasm for the historical profession.

12–13 (Good)

The essay includes thoughtful reasons why the author wants to study the history of mathematics but may lack originality. There are indications of growth in the author's sophistication in thought and expression, but at least one conceptual breakthrough is yet to be achieved. The author shows interest and enthusiasm for the historical profession and is developing a professional demeanor.

10–11 (Needs Improvement)

The essay is perfunctory in indicating the reasons why the author wants to study the history of mathematics. The author shows interest and enthusiasm for the historical profession. The complexities of historical practice are acknowledged but not resolved.

0–9 (Unacceptable)

The complexities of historical practice are misconstrued or ignored. The author's interest in the history of mathematics is not apparent.


Organization (10%)

9–10 (Excellent)

Thesis statement for entire essay is contained within first paragraph of paper, and all paragraphs have obvious topic sentences. Transitions between paragraphs make it very easy to follow the argument. Underlying logic of claims is apparent and sound. The reader understands the author's thinking throughout the paper.

8 (Good)

Thesis statement for entire essay is contained within first paragraph of paper, and all paragraphs contain topic sentences. Transitions between paragraphs are generally clear. Underlying logic of claims is sound, but reader may have to stop once to discern the author's thinking.

6–7 (Needs Improvement)

Thesis and topic sentences do not stand out to the reader. Paragraphs change topic abruptly. Underlying logic is difficult to follow. Reader has to read essay twice to understand it.

0–5 (Unacceptable)

There is no hierarchy among the sentences of the paper. It is impossible to follow the argument because the underlying logic is random or contradictory. There seem to be many arguments, and it is completely unclear which is the main one.


Clarity and Style (15%)

14–15 (Excellent)

Ideas expressed in active voice and clear, fluent sentences. All words are chosen for their precise meanings. All new or unusual terms are well-defined. Key concepts and theories are accurately and completely explained. No rhetorical questions or slang.

12–13 (Good)

Ideas usually expressed in active voice and clear sentences. Most words are chosen for their precise meanings. Most new or unusual terms are well-defined. Key concepts and theories are explained. No rhetorical questions or slang.

10–11 (Needs Improvement)

A few sentences are not clearly written. Passive voice is prevalent. Words are not chosen for their precise meanings. New or unusual terms are not well-defined. Key concepts and theories are not explained. Paper has several rhetorical questions or uses of slang.

0–9 (Unacceptable)

Prevalent problems with clarity. Essay uses words that do not fit the context at all. Phrasings are awkward and incoherent. Style reflects the author's stream-of-consciousness or conversational tone.


Mechanics (15%)

14–15 (Excellent)

Sentence structure, grammar, and expression are excellent; absolutely no run-on sentences or subject/verb disagreements. Correct use of punctuation. Minimal to no spelling errors.

12–13 (Good)

Sentence structure, grammar, and expression are strong despite occasional lapses. Punctuation is often used correctly. Some (minor) spelling errors; may have one run-on sentence, sentence fragment, or subject/verb disagreement.

10–11 (Needs Improvement)

Some problems in sentence structure, grammar, and expression, as well as errors in punctuation and spelling. The mistakes do not intrude on the reader's ability to understand the essay. May have several run-on sentences or fragments.

0–9 (Unacceptable)

Prevalent problems in sentence structure, grammar, and expression that interrupt the reader's concentration. Frequent (in nearly every sentence) major errors in punctuation and spelling. May have many run-on sentences, fragments, and subject/verb disagreements.


Citations & Formatting (10%)

9–10 (Excellent)

Specifed citation style (e.g., Chicago, MLA, APA) used consistently in footnotes and in bibliography; direct quotations, paraphrases, and summaries are all cited; paper meets any specified formatting requirements (e.g., for title, student name, font size, spacing, margins, page numbers, etc.); paper within the required length range.

8 (Good)

Specifed citation style (e.g., Chicago, MLA, APA) attempted in footnotes and in bibliography with a few minor inconsistencies between citations; direct quotations are all cited but author may have failed to cite 1 or 2 paraphrases or summaries; a few errors in any specified formatting requirements (title, student name, font size, spacing, margins, page numbers, etc.); paper within 10% of the required length range.

6–7 (Needs Improvement)

Specified citation style (e.g., Chicago, MLA, APA) attempted in footnotes and in bibliography with some substantial inconsistencies between citations; citations provided only for direct quotations; several errors in any specified formatting requirements (title, student name, font size, spacing, margins, page numbers, etc.); paper within 20% of the required length range.

0–5 (Unacceptable)

Frequent errors in specified citation style (e.g., Chicago, MLA, APA) that impede the reader's ability to identify and locate the sources used; multiple failures to follow any specified formatting guidelines; paper within 30% of the required length range.

HoM Toolbox, or Historiography and Methodology for Mathematicians: Introduction – Some Closing Thoughts

Amy Ackerberg-Hastings (MAA Convergence)


This article was designed to share the basics of historical practice with the wide range of people who might want to research and write the history of mathematics: instructors and students in history of mathematics courses; students contemplating graduate studies in mathematics and its history as well as academics and professionals considering a new direction in their careers; and experienced historians of mathematics looking for new ways to express and evaluate their own theoretical approaches to historical interpretation. The article should also provide benefits to those whose main goal is to become more knowledgeable readers of the history of mathematics or audience members in history of mathematics conference sessions.

We have asked a number of questions along the way, including:

  1. What is history? Why should we want to research and write it well?
  2. How do we know about the past?
  3. How do we create history based on what we know about the past?
  4. What is the history of the history of mathematics?
  5. How can we articulate our own philosophies of the history of mathematics?

None of these questions have a single correct answer, although some possible responses are supported by a stronger body of evidence than others. One thing all five questions have in common is that they require us to identify methods, techniques, and approaches essential to understanding how and why the past unfolded in the way that it has—the “tools” of the history of mathematics. Although your metaphorical toolbox should be much fuller as a result of reading this article, there will always be space for adding more tools via the many theoretical approaches to historical interpretation that are applicable to the broad, global discipline that is the history of mathematics.

HoM Toolbox, or Historiography and Methodology for Mathematicians: Introduction – References

Amy Ackerberg-Hastings (MAA Convergence)


Ackerberg-Hastings, Amy. 2019, April. Teaching Mathematics with Ephemera: John Playfair's Course Outline for Practical Mathematics. Convergence.

Ackerberg-Hastings, Amy, and Amy Shell-Gellasch. 2014, December. Online Museum Collections in the Mathematics Classroom. Convergence.

American Historical Association. 2019. Statement on Standards of Professional Conduct. https://www.historians.org/jobs-and-professional-development/statements-standards-and-guidelines-of-the-discipline/statement-on-standards-of-professional-conduct.

Barrow-Green, June, Jeremy Gray, and Robin Wilson. 2019–2022. The History of Mathematics: A Source-Based Approach. 2 vol. AMS/MAA Textbooks, vol. 45. Providence, RI: MAA Press.

Booth, Wayne C., Gregory G. Colomb, and Joseph M. Williams. 2016. The Craft of Research. 4th ed. Chicago and London: The University of Chicago Press.

Brown, Elspeth H. n.d. Writing About History. University of Toronto. advice.writing.utoronto.ca/types-of-writing/history/.

Conolly-Smith, Peter. 2007. Writing on History. https://qcpages.qc.cuny.edu/Writing/history/index.html.

Cronon, William, ed. 2008–2009. Learning to Do Historical Research: A Primer for Environmental Historians and Others. http://www.williamcronon.net/researching/.

Dauben, Joseph W., and Christoph J. Scriba, eds. 2002. Writing the History of Mathematics: Its Historical Development. Science Networks – Historical Studies, vol. 27. Basel: Birkhäuser Verlag.

Drake, Frederick D., and Sarah Drake Brown. 2003. A Systematic Approach to Improve Students’ Historical Thinking. The History Teacher 36(4): 465–489.

Grattan-Guinness, Ivor. 1994. Talepiece: The History of Mathematics and Its Own History. In Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, edited by Ivor Grattan-Guinness, ii:1665–1675. London and New York: Routledge.

Grattan-Guinness, Ivor. 2004. History or Heritage? An Important Distinction in Mathematics and for Mathematics Education. The American Mathematical Monthly 111(1): 1–12.

Green, Anna, and Kathleen Troup, eds. 2016. The Houses of History: A Critical Reader in Twentieth-century History and Theory. 2nd ed. Manchester, England: Manchester University Press.

Mandell, Nikki, and Bobbie Malone. 2007. Thinking Like a Historian: Rethinking History Instruction. Madison: Wisconsin Historical Society Press.

Mehrtens, Herbert, Henk Bos, and Ivo Schneider, eds. 1981. Social History of Nineteenth Century Mathematics. Boston: Birkhäuser.

Rael, Patrick. 2004. Reading, Writing, and Researching for History: A Guide for College Students. Bowdoin College. https://faculty.etsu.edu/fritzs/picts/Rael_Reading,%20Writing,%20Researching%20History.pdf.

Remmert, Volker R., Martina R. Schneider, and Henrik Kragh Sørensen, eds. 2016. Historiography of Mathematics in the 19th and 20th Centuries. Cham, Switzerland: Springer International Publishing.

Schrag, Zachary. 2007–2022. HistoryProfessor.org: Guidelines for History Students. George Mason University. https://historyprofessor.org/about-2/.

Spiegel, Gabrielle M. 2009. The Task of the Historian. American Historical Review 114(1): 1–15.

Stedall, Jacqueline. 2012. The History of Mathematics: A Very Short Introduction. Oxford: Oxford University Press.

Storey, William Kelleher. 2020. Writing History: A Guide for Students. 6th ed. New York: Oxford University Press.

Tosh, John. 2021. The Pursuit of History. 5th ed. London: Routledge.

Tosh, John, ed. 2018. Historians on History. 3rd ed. London: Routledge.

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

HoM Toolbox, or Historiography and Methodology for Mathematicians: Introduction – Acknowledgements and About the Author

Amy Ackerberg-Hastings (MAA Convergence)

I am thankful for fruitful discussions with online audiences at the 2021 Canadian Mathematical Society Winter Meeting, the 2022 Joint Mathematics Meetings, and the December 2022 Pennsylvania Area Seminar on the History of Mathematics (PASHoM), as well as for feedback from Janet Heine Barnett and the Convergence editorial board.

About the Author

Amy Ackerberg-Hastings is the co-editor of MAA Convergence and the CSHPM Notes column in Notes of the Canadian Mathematical Society. She taught historical research and writing skills to undergraduate history majors at University of Maryland University College (now University of Maryland Global Campus) for over a decade, supervising over 200 senior theses. In 2010, she and Jeff Glasco wrote modules and online course materials for expanding the former capstone course, Introduction to Historical Writing, into the current three-course sequence: Historical Methods, Historical Writing, and Senior Thesis in History.