A Writing Intensive General Education History of Mathematics Course

Table of Contents

Introduction

History of Elementary Mathematics Course

Course Design

Assignments

• One-page papers
• Research papers
• Final project and presentation

Conclusion

References

About the Author

Appendices

• A. Examples of one-page paper topics
• B. Calendar paper and poster details
• C. Student Escher "fan art" project
• D. Final project instructions

Introduction

Over the past few decades, how to have non-STEM students fulfill a mathematical graduation requirement has (finally) become a topic of conversation and source for new genres of courses and texts. Prior to this “revolution,” the vast majority of undergraduate students had to take a College Algebra course, even if they were never going to take another mathematics course. I feel this is akin to requiring all students to take at least one semester of a foreign language (which is not a bad idea in itself), but making that first course be only on conjugating verbs. Courses titled Finite Mathematics (beginning as early as the 1950s) and the generally more accessible Mathematics for Liberal Arts (beginning as early as the 1960s) constituted a gradual shift to classes in which students in non-technical majors could be more engaged and successful. Though these courses address the problem of using algebra as a terminal course for non-STEM majors, they have had varying degrees of success at addressing the issue of mathematics being a barrier to graduation for many students. Fortunately, many post-secondary institutions have moved past this limited approach to the one-mathematics-course minimum requirement. These institutions now offer one or more recently developed quantitative-literacy or general-education mathematics courses from a long list of overview as well as topical courses, such as “the mathematics of personal finance” or “the mathematics of sports.” Another alternative is a History of Elementary Mathematics course with a strong writing component.

History of Elementary Mathematics Course

The audience for general education courses are usually those who are not entering the STEM fields (Science, Technology, Engineering and Mathematics), and who are often math-phobic. My goal in creating a general education mathematics course using the history of mathematics as the theme is to give these students at least one engaging (might I even say “fun”?) experience in mathematics before they sail away from the subject forever. Since these students have often steered clear of mathematics for as long as possible, they can struggle with just about any mathematical content. Since that mathematical content must be provided in order to qualify for the mathematical or quantitative literacy graduation requirement, I aim to deliver that content in a form and context as appealing as possible.  This is done using a two-pronged approach. First, the history of mathematics is the perfect vehicle since students in general education courses often have wildly different backgrounds and abilities. Some students have very limited mathematical skills, while others are capable mathematics students but have never had an interest in the subject. So choosing just about any mathematical content will find some students having already experienced some of the material while leaving others behind. The history of elementary mathematics is (for the most part) new to all the students. It also has the advantage that they have all seen many of the topics already (theoretically). So they can hone their previous knowledge, while learning the historic, cultural, and logical underpinning of these topics. This can be especially interesting to them since many people wonder where some of the methods they were taught in school came from.

The second line of attack is incorporating several writing assignments, as well as projects and hands-on activities, into the course. General education students are often more comfortable in the realm of humanities or the arts. Writing assignments and projects allow these students to leverage their strengths in the process of conquering mathematics. This gives them a sense of confidence and competence from the start, which they may not feel when entering a traditional mathematics course. For the students who are less comfortable using their verbal skills, this course helps strengthen those skills. In either case, this course is clearly well placed to fulfill any “writing across the curriculum” goals you may have. The course described here is also very useful to pre-service elementary education majors.

Students benefit in several ways from approaching mathematics that they likely have seen before through a perspective that is new to them. First and foremost, the framework allows them to revisit the material outside of a remedial setting. Second, looking at the original context in which the mathematics was developed is often a much more holistic and intuitive approach than the polished and formulaic presentation of modern texts. An example of such mathematics is the method of false position and double false position for solving linear equations. In this problem-solving technique from the late Middle Ages, a “guess” is used that works easily with the problem, and then proportional reasoning is used to scale the guess up or down to arrive at the correct answer. I have found over the years that students who are not comfortable with algebra often intuitively use ad hoc methods that are very similar to this. If students are shown this holistic approach first, they can transition to the algebraic reasoning, and then more readily extend their understanding to the algebraic solution found in modern texts. The students then can solve the problem algebraically as well as gain a stronger sense of how one solves problems in general. This likewise provides them affirmation of their intuitive approach and their inherent mathematical ability. In addition to the historical methods mentioned above, hands-on projects can be used in class, such as making or using an abacus while discussing the different types of abaci, or making and using Napier’s Bones to showcase the lattice method of multiplication. Bringing such objects or models into the classroom adds to the historical as well as the mathematical experience for the students.

Course Design

As mentioned above, an historical approach can be used to leverage students’ humanities and writing skills. This is possible through the use of assignments that fit naturally into these non-STEM courses: discussions, readings, papers, presentations, projects, videos, performances, essay exams, and so on. Of course, mathematical homework and exam problems would still be assigned, but they can comprise a smaller portion of the graded assignments and in-class presentations.

The topics included in this course are chosen to allow students to see the wider context of the mathematics and the cultural influences involved, which is now a requirement of many college curricula. Included is a fair amount of traditional (non-mathematical) history and discussion showcasing the interplay of tradition, religion, and societal beliefs that shape and often guide mathematical and scientific advances. The text I use for this course, Algebra in Context: Introductory Algebra from Origins to Applications, published by the Johns Hopkins University Press in 2015 [3], was specifically designed to be used in such a course. Its full table of contents can be found at https://jhupbooks.press.jhu.edu/content/algebra-context under "Preview."

Course topics include:

• number systems and number bases of other cultures, leading up to the Indo-Arabic decimal place value system, including some systems of fractions
• calendars
• evolution of mathematical notation
• methods from various times and cultures for arithmetic calculations (for instance, Greek, Babylonian, Chinese, European methods of multiplication, division, and root extraction)
• solutions of polynomial equations up to the cubic
• exponentiation and logarithms (depending on the level of the students)
• Greek number theory of figurate numbers
• set theory.

I use a large dose of object-based learning in this course, and these objects are used in many instances as the prompts for the shorter writing assignments. Many of the images I use come from the Convergence Mathematical Treasures Collection and the Smithsonian Learning Lab online digital database and educational platform (www.learninglab.si.edu).

Assignments

Grades are based on a mix of traditional mathematical assignments such as homework and exams, writing assignments, and projects and presentations. Ten one-page papers, two four-page research papers, and a final project constitute the non-mathematical assignments most semesters I teach the course. But each semester I add or delete assignments based on student feedback. I encourage student input in the course. Since this is a terminal course, there is no reason to get pedantic about what is and is not included. The first several times I taught it, I let students pick the final two topics we covered from the text (via sticky note voting). They usually chose the same topics, so I now usually include those topics. The students did seem to appreciate the chance to choose, and I will continue to give them that option when appropriate. I give students the option to rework all homework assignments and short papers. Again, since this is a terminal course for non-majors, my prime objective is that they learn the material. As long as they turned in an assignment on time originally, they can get all the points.

One-page papers

Each short paper is on a specific topic of immediate bearing on the content. My favorite topics are:

• The Ishango bone as one of the first records of mathematical writing; alternatively, the Ishango bone compared to the Lebombo bone, an ancient ceremonial object
• The evolution of the Roman numerals (from the Etruscans)
• The oldest ancestor of our zero recently rediscovered in Cambodia
• The French Revolutionary Clock, or the French Revolutionary Calendar
• The competition between the Japanese abacus and the American adding machine just after World War II
• The history of our use of “x” in algebra, often a summary of Terry Moore's TED talk, Why ‘x’ is the Unknown [1]
• A review of a mathematical object group on the Smithsonian National Museum of American History website
• A review of an expository historical article I provide based on the topic at hand. I often use articles from Math Horizons since these are short and written at the appropriate level.
• A review of a news article on a recent mathematical or history of mathematics discovery (largest prime found, a new artifact, etc.).

Each of these topics contains an “aha!” moment that makes them fun to research (as long as the student digs deep enough to find it) and something that they might remember after leaving the course. Each topic also generates discussion in class about how mathematics is done in other cultures and how our assumptions can influence our views. Each paper must have two sources and an image. See Appendix A for three of my favorite examples: British tally sticks, the rediscovery of the “Cambodian Zero,” and the 1946 Japan/abacus versus U.S./adding machine competition.

There are several research pitfalls I know the students will make, and I use sources they easily find on the web to have short discussions after each of the first few papers are turned in. For example, an easy-to-find reference for the Ishango bone is part of a new-age aliens-built-the-pyramids type of website. But when students find the site via a Google search, it looks very reputable. Only when you back out of the URL do you find the bizarre material. We discuss:

• trustworthy websites and how to tell if you might trust a site
• how to cite your work, including images (Wikimedia is not the owner of its images; students need to track down the actual source)
• plagiarism.

Given all this, I grade the short papers more leniently at the start of the semester. I also allow students to rewrite any paper to address major issues, as long as they have touched base with me about what needs to be corrected. Again, my goal is that they learn the material and experience success.

Research papers

These two papers are short research papers that must have at least four sources (at least two print: journal or book) and include either a poster presentation in class or a class gallery tour (to be explained below). The two topics are:

• The history and general attributes of various calendars (Gregorian, Julian, Hebrew, Aztec, Muslim, B’hai, etc.). Mapping Time by Richards [2] is a great resource which I put on reserve in the library. The day the papers are due each student either (a) brings a full-size poster and presents a 4-minute overview of their research, or (b) brings a small poster about their topic and we conduct a “gallery walk” so all the students can be introduced to the various calendars, but no presentations. I decide which of these two options to use based on the amount of time I have that semester and how large the class is. Presentations can take a full class period or more, while the gallery walk with a short discussion following can be done in 30 minutes. See Appendix B for details, topic list, and grading sheet.
• Biography of a mathematician. I provide a short list of mathematicians related to the course, including some who are primarily scientists and several from minority groups. Students are also welcome to choose their own personage. I find that each semester, the female and minority students are more likely to choose their own subject from a cultural or ethnic group they identify with. Often this is someone they have heard about or found online by a search. Like the calendar paper, a short presentation or gallery walk of small posters takes place in class after the papers are due (time permitting of course). If I assign both of these longer writing projects, I often do fewer one-page papers and/or reduce the scope of the final project.

Part of each student's grade is to trade their draft paper with a partner (chosen in advance – I pass around a pairs sign-up sheet). I allow time in class about a week before the paper is due for partners to meet and give each other feedback. If more than one student does the same topic, I want them to pair with a student who did a different topic. They then turn in their marked-up draft so I can see that they really did comment on their partner’s work and got comments on their work from their partner. I make this worth 10/100 points. I feel it is important that students learn to proofread and how to get and give input and put it to use. This activity also helps stop students from throwing their papers together the night before.

Final project and presentation

Every student must do a final project on any topic related to mathematics or the mathematical sciences that they are interested in, not necessarily historical. Over the years, students have produced truly inspiring works from video documentaries, to works of art or music, to models of mathematical items, to traditional research papers on a wide range of topics. The only requirements are that they:

• Meet with me ahead of time to vet their idea. Since each student can pick their own topic and mode of presentation, I meet with each student to make sure that no one is doing too easy of a project or too hard of a project.
• Choose a topic that has some mathematical content.
• Hand in some sort of write-up that summarizes their project, how they proceeded, and their sources. For example, a student who liked to make jewelry made several items that incorporated the Fibonacci numbers. For the presentation, she brought the items, handed in a two-page summary of how the Fibonacci numbers showed up in each piece, and gave a short Power Point presentation to the class that included an introduction to the Fibonacci numbers and images of her creating the pieces.
• Include at least a little history, which can be simply a short summary of the history of whatever mathematical or scientific subject they chose.

During our last course meetings and our final exam period, students give their presentations. On all presentation days, I suggest we have a simple potluck. I want these days to be celebrations of their discoveries, not stressful “I have to give a presentation in class!” days. See Appendix C for a lovely example of a non-traditional final project. See Appendix D for the very simple assignment description I hand out. For both the longer papers and the final project, I require students to turn in a half sheet of paper (at the bottom of the assignment sheet) that gives their name, topic, and at least two references. This is due about two weeks prior to the due date of the assignment. This helps assure they get started early and have picked a topic and references that are appropriate. Of course, some don’t turn this in (it is worth 5/100 points) or do it cursorily and still wait to the last minute to start.

Conclusion

I have taught versions of this course at Beloit College, a small liberal arts college in Beloit WI, and Montgomery College, a large public two-year college in Rockville MD. As described above, the audience is non-STEM majors who are required to take one general education mathematics course to graduate (so I do get a fair number of seniors who have put it off!). Given that these schools, and most schools I would guess, have other options for fulfilling a general education mathematics requirement, they have chosen this course because something about it made them curious. I also make sure the advising staff know that this course is a great option for students who struggle with mathematics and have a strong interest in the humanities. In fact, this is the selling point of, and the main reason why, this course and the accompanying text book were developed. These students often would struggle through a more traditional course in the general education/quantitative reasoning category. There is no reason any student should fall through the cracks and be held up in their career goals for lack of an interesting and creative mathematics course. This does not mean that everyone passed my class with flying colors. But my experience has been that those who did poorly did so for lack of effort; in particular, not attending class.

The student response has been overwhelmingly positive. They appreciated the more diverse approach, both in the content and the assignment styles. They enjoyed learning interesting mathematics and history that related in some way to the mathematics they had learned in school. And the final project, where they had so much control of the topic they researched, left them with a positive experience. They left class excited about what they had researched and the knowledge that they were now an “expert” in at least one mathematics related topic. To end with an anecdote, one non-traditional student who took the course while at Montgomery College saved her textbook because she wanted both of her children (in high school at the time) to take the course when they got to Montgomery College. Teaching this course is great fun and very rewarding. I strive to make the whole course as communal and flexible as possible.

References

[1] Moore, T., Why ‘x’ is the Unknown, TED talk, 2012. https://www.ted.com/talks/terry_moore_why_is_x_the_unknown

[2] Richards, E. G., Mapping Time: The Calendar and Its History, Oxford University Press, 2000.

[3] Shell-Gellasch, A., and J.B. Thoo, Algebra in Context: From Origins to Applications, Johns Hopkins University Press, 2015.

About the Author

Amy Shell-Gellasch is a lecturer at Eastern Michigan University. She conducted research on historical mathematical devices at the Smithsonian National Museum of American History and currently creates educational resources and conducts faculty development workshops on the Smithsonian Learning Lab online educator platform for object-based learning. She is co-founder and chair of the History of Mathematics Special Interest Group of the Mathematical Association of America.

Appendix A – Examples of one-page paper topics

Students must include at least one correctly cited image for each paper.

British Tally Sticks

The British tally sticks were used by the British Exchequer starting in the 12^th century to record payments to the government such as taxes. A piece of wood would be marked with the payor’s name and the amount. The stick would be split lengthwise, the individual getting the “short end of the stick,” called the foil, while the government kept the longer piece, called the stock. These were then sold to investors. This is the genesis of our modern Stock Market. I love the words and phrases we use now that originated with this device. The story is even more fun if the student digs deeper. After hundreds of years, the government had too many tally sticks and in 1834 decided to burn them. The fires got out of hand and resulted in the Parliament Building burning to the ground. See the Convergence article Mathematical Treasures – English Tally Sticks, by Frank Swetz and Victor Katz, for many images of tally sticks.

Cambodian Zero

Recently Amir Aczel tracked down a lost engraving with an earlier than previously accepted version of the Indo-Arabic zero. The engraving is from Angkor Wat in Cambodia. Since the first unit of this course is various number systems, I like this assignment because it is a recent rediscovery. Aczel’s delightful book, Finding Zero, is easy and fun to read, and I suggest that students read and write a review of it for extra credit. Information on the discovery can be found on the Smithsonian Magazine site: https://www.smithsonianmag.com/history/origin-number-zero-180953392/ or see the Convergence article, Mathematical Treasure: The Cambodian Zero.

Abacus vs. Adding Machine

One of my favorite historical mathematical events is the competition in 1946 (yes, that soon after the war) of a Japanese abacus versus an American electric adding machine. Guess who won?! I like the students to find an image of an electric adding machine of the era. The U.S. Army used Burroughs adding machines. Images and information on early adding machines (along with many other mathematical devices) can be found at the Smithsonian Museum of American History’s Object Groups collection: americanhistory.si.edu/collections/object-groups/adding-machines

I also show at least one adding machine during class so we can have a discussion of electric versus electronic devices. Today it is hard to find anything that is not electronic (has a circuit board) in the classroom. Even the light switch is probably electronic if it has a motion sensor. I also like to point out that the adding machine is mechanical with a paper readout, and thus pretty slow. Pictured below left is a Burroughs’s Class 9a ca. 1944 adding machine and below right a cut-away image of a Class 8 ca. 1925 (Smithsonian Learning Lab database) adding machine to show the inner workings.

One of the best sources on this competition is the 2009 Wired Magazine article, Nov 12, 1946: The Abacus Proves Its Might. Or see the more detailed excerpt from the book, The Japanese Abacus: Its Uses and Theory, by Takashi Kojima.

Appendix B - Calendar paper and poster assignment

Download instructions, topics sign-up sheet, and grading outline (PDF) (Word) for this assignment. Though many different calendars have been used throughout human history, finding reference material on some of the lesser known ones is hard. So the sign-up sheet has space for two students to sign up to do the same topic (but not work together). 

Appendix C – Example of student project: Escher 'fan art'

In the fall of 2016 I had an art major by the name of Caroline Kraegel in my course at Montgomery College. She decided to do "fan art" based on M. C. Escher’s tessellation paintings. She got so involved in the project that she regularly gave me updates and said that she ended up spending “way too much time on it.” But she clearly found herself more and more interested in the topic. (She had little prior knowledge of Escher’s work.) She created three paintings which she let me keep! (She stated that they were not her best work, but I think they are wonderful. As a side note, she also worked at an Italian bakery making hand-made doughnuts. So, on the presentation days she brought individualized doughnuts for the class, in a box decorated with “I love math” on it. This from a student who self-identified as math phobic the first week of class!) She turned in a two-page paper that gave a short biography of Escher, a brief description of the geometry he used – to the best of her ability, and her list of sources. Her presentation briefly mirrored her paper yet focused on the paintings she created and the mathematics in them as well as images of the Escher originals she used as inspiration. Can you figure out which Escher works she used?

Appendix D - Final project instructions

Download Final Project Instructions (PDF) (Word).

Editor's note: For descriptions of many more history of mathematics courses at all levels, see the upcoming MAA volume, The Courses of History, edited by Richard Jardine and Amy Shell-Gellasch, to be published during 2018 in the MAA Notes series. Volumes in this series are available free online to MAA members.