Fourteen HOM SIGMAA Undergraduate Student Paper Contest wins over fifteen years, 2004–2018—I am as surprised as you are. Students from my once-a-year, little History of Mathematics class (mean enrollment 12, median 13) at the University of Missouri–Kansas City (UMKC) have now won or co-won first place eleven times and runner-up/second place three times. In 2004, the first HOM SIGMAA contest excited us. Undergraduate expository papers in any context, but especially in mathematics, rarely find contest or publication opportunities. Knowing the HOM SIGMAA contest exists has motivated my students year after year to write beyond what is strictly necessary for a grade, and often beyond their own expectations. We have submitted papers to the contest every year since 2004. And although we did not win every year, enduring one notable three-year dry spell early on, my students’ record of success is, frankly, wonderful. You’ll read the list of winning papers at the end of this article. So, it is worth asking, “How did we get here?”
First you should know that although I am the instructor of this course, my training is in mathematics and not in the history of mathematics or the history of science. In those areas I am an autodidact. Second, as a teacher of writing I am also an amateur, though over twenty years I have become more skilled as an editor and writing mentor. So, the truth is that I stepped into the teaching of a History of Mathematics course at UMKC in 1998 untutored but with an enthusiasm for its importance in the training of our mathematics majors. Here follow a handful of my idiosyncratic observations and hand-crafted techniques in a distillation of what I’ve absorbed from historians and others over time. I hope these observations offer help to any teachers of the history of mathematics and their students inspired by and considering submitting papers to the HOM SIGMAA contest.
In the spring of 1998, I volunteered to teach our History of Mathematics course, which, although it had a senior level number, had no prerequisite, and had only been taught on and off over the years whenever a faculty member showed interest at that moment. Often students with little or no background in mathematics enrolled. So, I decided immediately that a mathematics prerequisite was needed, that it should be taught as what our campus called a “writing intensive” course, and that it should be offered once a year, every year.
Writing intensive (WI) courses at UMKC now include the following relevant goals, which I helped compose. At the end of the course students should be able to:
Demonstrate their ability through writing to read closely and analyze critically the texts of their disciplines.
Produce writing through the recursive process of brainstorming, research, drafting, peer review, and revising.
Use research methods and documentation that meet the standards of the discipline.
Articulate and discuss their work with peers or the instructor.
WI course construction requirements include:
The course design emphasizes and teaches writing as a recursive process.
(The recursive process is defined as submission of one or more preliminary drafts for instructor response; peer review; revision of content, form, mechanics, and style, leading to a final draft.)
Writing assignments are distributed throughout the semester and differ in length and purpose.
The course requires a total of 5,000–10,000 words of revised, final-draft quality writing.
Writing assignments account for at least 40 percent of the course grade.
In spring 2000, the course became designated Math 464 WI, History of Mathematics, writing intensive. [For more details see the Math 464 WI - Writing Intensive Chart (pdf)]. Since a writing intensive course is required for graduation at UMKC, all undergraduate students have completed a couple of basic writing courses by the time they are juniors to prepare them for such a course. So, I would not have to teach basic writing skills. Soon after Math 464 WI became writing intensive, I added the prerequisite of our On Solid Ground: Sets and Proof course, which itself has a prerequisite of Calculus II. Following these changes, I had classes of junior or senior mathematics majors or minors trained to write reasonably well, as well as read and understand proofs. The course quickly evolved into an introduction to the history of mathematics through the study of about forty proofs. The required texts have always included A History of Mathematics: An Introduction by Victor J. Katz, second or third edition; Journey Through Genius: The Great Theorems of Mathematics by William Dunham; and others as I found them useful. For example, I now also require How to Read Historical Mathematics by Benjamin Wardhaugh. Except for one year when it was cancelled, we have offered and I have taught Math 464 WI once a year each spring since 2000.
Students write two 4,000–5,000 word papers for my course, and the first and most time-intensive task for them is to choose a topic for each paper. This is never simple. Their incoming knowledge of the history of mathematics is shallow at best. So, to complement the resources within the Katz textbook (his References, Exercises, and Notes), I provide a list of print and internet sources for them to survey. (See Appendix 2.) I also bring in a subject expert librarian for one class who teaches them how to navigate library resources. In addition to our UMKC library, we are lucky to have located in the center of our campus the well-known private Linda Hall Library of Science, Engineering, and Technology, containing not only a huge collection of books in the history of mathematics, and subscriptions to journals like Historia Mathematica, but a well-stocked rare book room that houses dozens of historical mathematics books. (See the Convergence article: Mathematical Treasures at the Linda Hall Library, by Cynthia J. Huffman.)
For the first paper, I allow my students at least three to four weeks to trap a topic, with my help and their extensive searches. (Yes, it often seems like netting a darting butterfly.) I tell them a topic must have original documents (in translation) available, and those must include enough argumentation or proof for a healthy and required explication. (It is always dangerous to choose a mathematician who writes too well. In that case there may be little for a student to add. Example: Euler.) Below, I will only concentrate on the unique and most mathematical part of these papers.
At the beginning of the course, for practice students complete a couple of stand-alone explications of documents provided by me. Then, once they begin work on their papers, similar but different explicated documents that they discover become the core of those papers. In concentrating on explication, I am tapping into our mathematics students’ aptitude and interest in precise, technical work, the nuts-and-bolts of mathematical tasks. As well, I am tapping into their problem-solving inclinations and imagination in the form of detective work to discover what the author wrote. I tell them they have both the advantage and disadvantage of hundreds and in some cases thousands of years of mathematical progress and training in their background beyond that of the author they are reading. The advantage is that they can often “see through” what the arguments are leading to, as they jot notes in the margins using modern symbolism and graphs, and so on. The disadvantage is that by engaging that modern perspective they entirely miss the cultural context of the time, and in fact fail to “see” what the writer has written. Their modern understanding eclipses their attention to the details of what is on the page. We work on stripping back their “overview” to discern what is said and what is not said, what is assumed in that time period and what is anachronistic and ahistorical.
Explications must be “just in time.” It is annoying and loses the reader’s attention to read a long passage which makes little sense to them, only to be followed by a long explication. Such an explication is too late, and often hard to precisely align with the points in the passage where it is meant to apply. It also tempts the reader to ignore the original passage and just read the explication, which denies them the true understanding of the context as well as the substance of the arguments. I will tell students “The author is the focus, not you. With the assistance of your gentle guidance, your light touch, we want to be able to read the author’s work without pause.” I share with my students what the Oxford English Dictionary writes:
Explicate, verb:
To explicate is to cleverly, subtly answer questions before they are asked, so the reader, when he or she raises these questions, realizes the answer has already been handily provided. Thus, apprehension of the material and moments of clarity arrive more quickly because the reader’s intellectual struggle has been diminished by the explicator's apt remarks. An explication is not provided in order to train readers; it is instead to ease their path to understanding what the author wrote. Examples of helpful insertions include:
In the beginning, students are not sensitive to where these additions may be needed. For instance, they often include long explanations of parts well-written by the author (as an unconscious act, I think, to prove to themselves that they understand), and neglect spots of confusion that they “saw through” but which desperately require explanation. My job is to generously red ink their early attempts and point out where there is real need for explanatory assistance. I remind them, “It is harder to be clear than you think!”
Here’s the sheet of advice I give my students for those short practice explications in advance of and separate from writing their papers, although of course the advice is universally useful. As I mentioned above, papers will then center on similar explications, but not consist solely of them. Note that their first task is to accurately type up the original source document, which seems easy, but often reveals a novice inattention to details like unusual or archaic spelling and capitalizations, grammatical oddities, and so on. This too turns out to be good practice for them.
Historical “Proof” Explication
When you read my Explications (and when you review your own)
I think it wise to provide examples to my students of what sort of explication product I expect. As I explain, I don’t require them to produce as complete a document as mine, but in reading my sample they should see what is possible with a little care and enough time on task. Every document requires multiple passes and close reading at both the level of fine detail and the level of structure and argumentation to reveal what explication is needed. A cursory glance (meaning just one quick draft) will inevitably miss important features, as I have found to my chagrin often enough in my own work. Here’s one of my samples. The original author’s document can be found here: Fibonacci On Two Birds Source (pdf). A second example document and my corresponding explication can be found in the following two documents: Alkhwarizmi Quadratic Equation Source (pdf) and Alkhwarizmi Quadratic Equation Explication (pdf).
One Explication SAMPLE provided to students
Math 464 WI History of Mathematics R. Delaware
A Transcription and Explication using Modern English and Notation
From: Leonardo Pisano (Fibonacci), c.1170–c.1240, born in Pisa, in (what is now) Italy, from Liber Abaci (Book of Calculation), 1202, Chapter 13, Part One, as translated by L. E. Sigler in Fibonacci's Liber Abaci: A Translation into Modern English of Leonardo Pisano’s Book of Calculation, Sources and Studies in the History of Mathematics and Physical Sciences, New York: Springer, 2002, pp. 462–463.
Notes: All comments in [square brackets] and illustrations are mine.
On Two Birds
Two birds were above the height of two towers; one tower was 40 paces in height and the other 30, and they [the towers] were 50 paces apart; at an instant the pair of birds descended [from the tops of the towers] flying to the center [some point between the towers, not necessarily the exact center] where there was a fountain, and they arrived at the same moment at the fountain which was between both towers. From the moment they left until the moment they arrived they flew in straight lines from the tops of the towers to the center of the fountain; the flights were of equal lengths [distances];
[The unstated question is: Where is the center of the fountain located? Now, we start on a solution.]
in geometry it is clearly demonstrated that the height of either tower multiplied by itself added to the distance from the tower to the center of the fountain multiplied by itself is the same as the straight line from the center of the fountain to the top of the tower multiplied by itself [This is a statement of the Pythagorean Theorem, Euclid I.47.];
this therefore known, you put it that the distance from the center of the fountain to the higher tower is any number of paces, we say 10,
[a first “false position” guess. Here he begins an application of the so-called “Method of Double False Position” (which my students have already studied; see the Note following this explication). We’ll discuss below whether this is appropriate.]
and you multiply the 10 by itself; there will be 100 that you add to the height of the higher tower multiplied by itself, namely to 1600; there will be 1700 that you keep [the result of his first application of the Pythagorean Theorem to the “higher tower” right triangle], and you multiply by itself the remaining distance, namely the 40 which is the distance from the center [of the fountain] to the lower tower; there will be 1600 which you add to the height of the lower tower multiplied by itself, namely 900; this makes 2500 [the result of his second application of the Pythagorean Theorem to the “lower tower” right triangle] that should be 1700 [because the bird flights (hypotenuses) were of “equal lengths”] as was the sum of the other two products; therefore [the value of] this position is long of the true value by 800, namely the difference between 1700 and 2500
[The value of “this position” is really the difference “lower tower sum of squares – higher tower sum of squares”, a function of the “position”, which here is 10 paces. But, since the bird flights (each a hypotenuse) were of “equal lengths”, the squares of those lengths are then also equal, so by the Pythagorean Theorem the “true value” of “lower tower sum of squares – higher tower sum of squares” must therefore be zero!];
therefore you lengthen the distance from the center of the fountain to the higher tower; indeed it is lengthened 5 paces from the first position, namely 15 paces [the second “false position” guess] from the center [of the fountain] to the higher tower, and you multiply the 15 by itself; there will be 225 which you add to height of the higher tower multiplied by itself, namely 1600; there will be 1825. Similarly you multiply by itself the 35 which is the distance from the center of the fountain to the lower tower making 1225 [corrected from typographic error “12225”]; this added to the 900, namely the height of the higher [should be “lower”] tower multiplied by itself, makes 2125 that should be 1825 by the abovewritten [sic] rule [again, because the bird flights (hypotenuses) were of “equal lengths”]. Therefore the value of the second position is an amount long of the true value by 300
[Once again, the “value of the second position” is really “lower tower sum of squares – higher tower sum of squares”, a function of the “second position”, here 15 paces. So, the “true value” is again zero.];
the first value was long indeed by 800; therefore you say:
for the five paces which we lengthened the distance from the center of the fountain to the higher tower we approximated more closely to the true value by 500 [800 – 300 = 500]; how much indeed shall we lengthen the distance from the center of the fountain to the same higher tower in order to improve the approximation by [an additional] 300?
You multiply the 5 by the 300, and you divide by 500; the quotient will be 3 paces [written within a small “answer” box below]
[The idea is that when \(\frac{5}{500}=\frac{\Delta}{300}\), it follows that \(\frac{5\cdot300}{500}= \Delta=3\). The large box above is meant to suggest, reading horizontally, that since a change of 500 was caused by an increase of 5 paces, then a change of 300 will be caused by an increase of 3 paces. The two *s on the main diagonal indicate the multiplication of “5 by the 300”.]
which added to the 15 paces yields 18 paces, and this will be the distance from the [center of the] fountain to the higher tower. Truly the remaining distance, namely the 32 [which is 50 – 18], is the distance to the lower tower.
[Finally, he checks the solution.]
For example, the product of the 18 by itself added to the product of the 40 by itself makes as much as the product of the 32 by itself added to the product of the 30 by itself, as had to be. [18^{2 }+ 40^{2} = 32^{2 }+ 30^{2} (= 1924).]
A Modern Direct Solution
Let \(x\) = the distance from the higher tower to the center of the fountain, so that
\(50 - x\) = the distance from the lower tower to the center of the fountain.
By the Pythagorean Theorem, since the bird flight lengths (each a hypotenuse) are equal, and hence the squares of those lengths are equal, the problem requires that
[Higher tower sum of squares] = [Lower tower sum of squares]:
\[\begin{array}{ccc} x^2 + 40^2 &=& (50-x)^2+30^2\\x^2 + 40^2 &=& 50^2 - 100x +x^2+30^2 \\ 100x + 40^2 -50^2 -30^2 &=&0 \\100x - 2\cdot 30^2 &=&0\\100(x-18)&=&0\\x&=&18\end{array} \]
So \(x= 18\) paces is the solution. Notice that it becomes clear at the third equation that in fact this is a linear equation (the “squared” powers add to zero) of the form \(ax = b\), justifying the use of the Method of Double False Position, which only applies to such linear relationships (functions).
Note: The Method of Double False Position
(called by Leonardo Pisano “elchataym”, an Arabic word)
[Based on comments by the translator L. E. Sigler of Leonardo Pisano’s Liber Abaci (Book of Calculation), 1202, translation published 2002, p. 628.]
Using modern notation, the idea is that to solve a linear equation \(ax+b=c\) for \(x\), pick two arbitrary values for \(x\), say \(x_1\) and \(x_2\) each of which is unlikely to be the correct value for \(x\), hence are in general called “false position” values. Substitute them into \(ax+b=c\) to obtain \(ax_1+b=c_1\) and \(ax_2+b=c_2\). From these two equations solve for both \(a\) and \(b\), then substitute those results back into \(ax+b=c\) , and finally solve for \(x\) to get:
\[ x=x_2+\frac{(c_1-c_2)(x_2-x_1)}{c_2-c_1}.\]
In “On Two Birds,” \(x_1 = 10\), \(x_2=15\), \(c_1=800\), \(c_2=300\), and \(c=0\), the “true value,” yielding the quotient
\[\Delta = \frac{(c_1-c_2)(x_2-x_1)}{c_2-c_1} = \frac{(0-300)(15-10)}{300-800}=\frac{-300(5)}{-500} = 3 ,\]
so that \(x = 15+3 = 18\), as desired. Also, in this modern notation, the proportion visualized in the “On Two Birds” box
\( \displaystyle{\frac{5}{300} = \frac{\Delta}{300}} \) is \( \displaystyle{\frac{x_2-x_1}{c_1-c_2} = \frac{\Delta}{c_2-c_1}} \)
matching exactly the calculation of \(\Delta\) above (and avoiding the appearance of any negative numbers, which Leonardo would not have used).
A Modern Interpretation of the Leonardo Pisano Solution
Using \(x\) and \( 50 – x\) as above, he examines the function of \(x\) (“the value of the position”) given by
\[ f(x) = \mbox{ [Lower tower sum of squares] -[Higher tower sum of squares]} \] meaning \[ f(x) = \left [ (50-x)^2 + 30^2 \right ] - \left [ x^2 + 40^2\right ] \]
which he desires to equal zero (the “true value”), that is, \(f(x) = 0\).
Using the Method of Double False Position (which does apply here since the function is actually linear), he first calculates
\( f(10) = 800 \) then \( f(10 + 5) = 300 = 800 – 500 \) ,
determining the points (10, 800) and (15, 300) lying on the desired line, and he finds the solution by extending that line until it meets the horizontal axis. That is, he concludes that if an increase (from 10 paces) of \(x\) by 5 paces yields a decrease of \( f(x) \) by 500, then a further increase of \(x\) by 3 paces will yield an additional decrease of \( f(x) \) by 300, meaning a decrease to zero. Thus \( x = 15 + 3 = 18\) paces is the solution.
By the way, the equation of that line is \( y = 1800 - 100 x = 100 (18-x) \).
Here’s the handout I provide students about paper construction. (Additional technical details are in Appendix 1.) Of course, it is in the many conversations we have throughout the semester and multiple revisions that all these points begin to make sense to them.
Notes on the Writing of Expository Mathematics Papers
I also include “Advice from Writing Experts” (see Appendix 1).
Because the contest spring submission deadline is mid-semester, of the two papers written for my course only students’ first papers are ready on that date. However, as the rules allow, occasionally we have submitted second papers from the previous spring class. In all fourteen wins, eight were first papers, and six were second papers; eight authors were male, six female. Six of the topics were chosen by students, each a welcome surprise for me, and eight were topics resulting from my guidance. The papers we submit to the contest are a subset of those submitted for the course assignment that earn positive reviews from me. Nevertheless, my personal contest win prediction record is poor! I’m never certain that my evaluation criteria for the course will produce a win—another healthy way the contest encourages better writing, as I learn from which papers win how to (perhaps) improve my teaching next time. We just submit three to six such papers each year and trust to the judges.
Finally, I always look for other outlets for well-written undergraduate expository papers. Thankfully, some of the submitted HOM SIGMAA contest papers and others written for the course have found print publication. [For a summary of my students' successes to date, see the table in UMKC-OtherPublications (pdf).]
At UMKC our English Department publishes annually The Sosland Journal, edited by English graduate students, that features essays by student winners of the Ilus W. Davis Writing Competition and other high quality papers: “The journal exhibits exemplary writing from both composition courses and university-wide writing intensive courses to be published and distributed to a larger audience, including UMKC students who use the journal as a textbook in select writing courses and the Kansas City community at large.” Since 2002, my students have been published there thirteen times and won eight awards.
In 2005, our UMKC Honors Program, now the Honors College, launched a journal called Lucerna, which is “interdisciplinary … and UMKC’s only journal … to cultivate and showcase original research and scholarship from the entire UMKC undergraduate community… peer-reviewed by a volunteer staff of student readers from the Honors College as well as faculty and staff from the Honors College.” My students have been published there eleven times since 2007.
In 2015, as I was looking for another publication opportunity for a couple of second student papers on more advanced topics, I discovered the Rose-Hulman Institute of Technology Undergraduate Mathematics Journal. Its description seemed promising, including (italics mine!): “devoted entirely to papers written by undergraduates on topics related to the mathematical sciences … each paper must be sponsored by a mathematician familiar with the student's work and … will be refereed. … Although the paper need not contain original research in mathematics, it must be interesting, well written, and at a level that is clearly beyond a typical homework assignment. Readers of the journal should expect to see new results, new and interesting proofs of old results, historical developments of a theorem or area of mathematics, or interesting applications of mathematics.” Two of my students’ papers survived the gauntlet of referees and were published there in 2016.
Finally now, the Honor Roll of UMKC HOM SIGMAA winners is…
[Editors' Note: All winning papers in the HOM SIGMAA Student Contest are published in Convergence; most are also available through the HOM SIGMAA archives at http://historyofmathematics.org/archive/.]
Richard Delaware has a Mathematics Ph.D. and is a Teaching Professor in the Department of Mathematics and Statistics at the University of Missouri – Kansas City (UMKC). In 2017, he won the UMKC Award for Excellence in Mentoring Undergraduate Researchers, Scholars, and Artists, in 2013 a small but meaningful Sosland Celebration Certificate of Achievement (for teaching writing) from the UMKC English Department, and in 2011 both the MAA Missouri Section Award for Distinguished College or University Teaching of Mathematics as well as the UMKC Provost’s Award for Excellence in Teaching. In 2018 he enjoyed the happy convergence of his background in theatre and his knowledge of the history of mathematics by creating copy for the mathematics posters used in the Kansas City Repertory Theatre’s production of “The Curious Incident of the Dog in the Night-time.”
The author thanks both editors and the three referees for sharp observations that substantially improved this article.
This appendix includes course requirements and advice on writing mathematics that I share with students in my course.
Basic Technical Paper Requirements in my Course
Advice from Writing Experts
From Strunk and White, The Elements of Style, 1972 edition:
The references below, such as [6, p. 10], are to the list of publications about writing mathematics that follow these bullet points:
References on Writing Mathematics
This Appendix includes information on researching the History of Mathematics which I share with students in my course.
Print Sources
Sources of Original Mathematical Documents in Translation
General Histories of Mathematics and References
Specific Problems or Theorems
Individual Mathematicians
Chinese and Japanese Mathematics History
Islamic Mathematics
Probability, Statistics, Graph Theory, Number Theory
High Quality Internet Resources
Many of these resources have links to other sites of interest. Be critical of the information you find!