By Jeffrey M. Ringer
December 6, 2006

At some point in the history of education, a line of demarcation was drawn between those who liked writing and those who liked math. Professors Sam Shore and Rita Hibschweiler, who both teach mathematics, are blurring that line.

One of the courses they teach, Math 531, “Mathematical Proof,” is a writing-intensive (WI) course billed as an “Introduction to reading and writing proofs in mathematics.” And 531 lives up to its billing—Hibschweiler and Shore make sure of that. In the past, Shore has required a term paper of his students, while Hibschweiler stresses the need for “clear, well organized writing” in her syllabus. Their courses, which meet four days a week, emphasize clarity as much as brevity, logic as much as writing.

Students in Shore’s course composed a paper “on the nature of a mathematical proof.” These terse statements commented on assumptions, theorems, axioms, proofs, properties, claims, and logic. One student averred that proofs require an unshakeable foundation in order to stand, analogizing mathematical proofs as a house of cards to make her point.

“The first layer has to be extremely strong to be able to hold the rest of the cards,” she wrote. “Only after we complete this layer are we able to build up another one”—and only after students have written their original paper can they critique their own ideas. Writing specialists describe this return to earlier writing “recursive.”

Like any good writing teachers, Shore and Hibschweiler stress the importance of audience. In critiquing their term papers, Shore instructs his students to ask, “What knowledge have you assumed on the part of your reader?” And his students follow through.

One noticed he had “assumed the reader would accept” commonly-known mathematical principles as “fact.” Another student conjectured that “the reader of the paper would be familiar” with the terminology of mathematical proofs, guessing that his audience consisted of his peers. He was right. Shore notes in the instructions for the assignment that it ought “to be written for other members of our class…”

Hibschweiler reserves one day a week for working through problems with her students and seems to always find a way to relate audience to every point she makes. She opened a recent Thursday problem-solving session by gently rebuking her students.

“You had the ideas right in your homework,” she said, “but you didn’t present them right. And that’s a critical part of this course.”

Critical, indeed. And if the students had any doubt concerning her word choice, they soon discovered its veracity. Working from right to left, Hibschweiler filled the whiteboard with statements, axioms, theorems, working towards a proof.

“What just happened here?” she asked, motioning to a series of statements on the left side of the board. “What did we just accomplish? Let’s tell our readers.” She moved to the right side of the board. “This told them we were going to do something; this told them what we were going to do. Somehow you need to let me know—or the person reading know—that here” (she motioned to another part of the board) “you’re using two steps. So we’ll write, ‘First, let n equal 1,’ followed by, ‘Then, the function of x equals the function of one.’ Here,” (yet another part of the board) “we could say ‘on the other hand’ or we could say ‘also.’ The only time we need to use English is when we need to announce that we’re moving on to something else.”

The language of mathematical proof differs from quotidian English in that words in mathematics are used more precisely.

“In math, common English words such as ‘and’ and ‘or’ take on different meanings,” said Kirsten Hutchison, Sam Shore’s undergraduate writing fellow.

Caroline Gumpert, Hibschweiler’s writing fellow, concurs. “With math, there is a much more inherent right and wrong to everything you write. Although there is variability, the end result must always be the answer. Your proof either works or it doesn’t.”

A writing fellow is a student chosen by the professor of the student’s particular course and was a strong writer for that course. The writing fellow is almost like a writing TA that helps provide the current students with guidance on how to write effectively for the class.

Hutchison remarks that “Math 531 is based on writing that is not found in your typical writing class,” and a quick glance at one statement of this “math language” corroborates her point—many of us wouldn’t understand how exactly to decipher the theorem for basic properties of set equality.

But that’s exactly why Gumpert and Hutchison interact with 531 students—as former students themselves, they learned to translate from English to math, and they can use that knowledge to help their peers, an undertaking they both find rewarding.

“As a fellow, you have to step back and dissect the information in order to clarify it for students because they might have different expectations than the teacher,” said Gumpert.

The translation necessary in Math 531 invariably involves that from English to math, but, equally as important, it involves the translation undergraduate writing fellows can help to achieve—that from teacher to peer.