# Writing About Math

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When course requirements at the Massachusetts Institute of Technology shifted 10 years ago, faculty members in the mathematics department found themselves with a new task in their job description. Not only did they have to teach their students to solve equations; they also had to instruct them in writing and communicating effectively on the subject.

This change in duties -- which mirrored similar shifts in the teaching of discipline-specific writing at other institutions -- gave rise to a host of new challenges, from the administrative to the pedagogic, said Haynes Miller, professor of math at MIT. The math faculty there had to learn how to teach the subject from a different perspective -- one in which words, not just numbers and symbols, are given emphasis.

Arriving at a common definition of effective teaching of writing and communication courses proved to be another obstacle, said Miller, because these classes were taught in seminars, many of which followed unique syllabuses or reflected the preferences and styles of the faculty and students in the courses.

Now, a decade later, members of MIT’s faculty have been developing -- with money from the National Science Foundation -- a website, the Educational Collaboration Space, that is meant to be a forum for those teaching communication skills to mathematics students. Faculty will be able to crowdsource their ideas, post lessons, exercises and classroom examples, reflect on their experiences, and develop some consensus about what works.

“I believe it will be an enormously helpful resource,” said Mia Minnes, SE Warschawski Visiting Assistant Professor in the math department of the University of California at San Diego (she also worked on early iterations of the project at MIT while an instructor there). “Teaching these skills is challenging in any context, and even more so for instructors whose background and training is in mathematics rather than communication.”

MIT hopes the web-based tool that its faculty has devised for teaching writing about math will have a wide application to other disciplines or universities. “It has nothing to do writing itself or mathematics. It could be anything,” said Miller. “Here’s a mechanism for them to form a community, and it’s much more immediate than having department meetings.”

This is far from the first time that someone has thought to develop a platform for crowdsourcing ideas about teaching -- either generally or about math. Members of many a math faculty have used wiki pages for this purpose. The Mathematics Association of America has its own digital library of teaching ideas on the subject (the MMA, which is a co-recipient of the NSF grant with MIT, will eventually host the completed site that will be dedicated to teaching communication skills to math students). The effort at MIT, which is still under development, is intended to be a user-friendly platform for teachers. Still, some expressed skepticism that the diffuse nature of the web would ever lend itself to one site's becoming the authoritative repository for anything, as some behind the MIT effort are hoping will be the case.

**Mathematicians' Motives**

The idea that students of math should be able to express their ideas in words is at least decades old, though the goal might more often be proclaimed than fully realized. The underlying motivation for wanting to teach math students to write about their subject also varies among the subset of mathematicians interested in the concept.

Some favor teaching aspiring mathematicians how to write because they feel it facilitates learning. “Clarity of expression is connected with clarity of thought,” said Miller of MIT. “You can’t write something clearly unless you understand it. You can’t understand it clearly until you write it.”

Writing can be a particularly useful tool when math students make the transition from computationally focused courses to more theoretical ones, said John Meier, professor of mathematics at Lafayette College, and a co-author (with Thomas Rishel) of *Writing in the Teaching and Learning of Mathematics*. "The use of writing assignments in courses in the calculus sequence is a great way for them to begin to see that there are ideas and logical structure underpinning the computations," he said.

Others see good mathematical writing as an end in itself. According to this view, mathematicians need to do a better job of writing clearly about the subject when addressing readers -- both within and outside the discipline.

"There’s long been a feeling among mathematicians that we don’t do a good job of communicating to each other and the outside world,” said Stephen B. Maurer, professor of mathematics at Swarthmore College -- whose own thoughts on the subject date to at least 1991, when he wrote an article in *PRIMUS*, a journal dedicated to the teaching of mathematics to undergraduates. "Mathematicians have a hard time reading other mathematicians. Everything you can do to make it clearer is valuable."

Minnes of San Diego said this lack of clarity can have serious consequences. “Mathematics has proven time and again to be the key to many important achievements in science and engineering,” she said. “However, these achievements typically require collaboration of many people over many years to mature; such collaboration depends on the articulation of technical ideas among the people involved.”

While every discipline has its own codes of language, styles and conventions, math allows writers to explicitly define terms (as in the phrase “let x=4”). These and other facets make the characteristics of good mathematical writing different from generally accepted ideas about good writing on a general subject, said Maurer.

For example, he said, “elegant variation,” or the use of synonyms to add variety and shades of meaning, is frowned upon in mathematical writing. “That is exactly the opposite of what you should do in mathematics because we split hairs over meanings; if you use a different word, a mathematical reader assumes you meant something different and the difference is important," said Maurer. "Writing in general is difficult to do well, but in math it’s especially difficult."

Finding an instructor with the right blend of skills to teach mathematical writing also can be a challenge. While it is one thing to teach writing on a general subject, it is quite another to write about mathematics, said Susan Ruff, lecturer in the Writing Across the Curriculum office at MIT and, some say, the repository of much of the institution’s knowledge in teaching math students how to write about the subject.

Ruff described an assignment that she uses in her classes, which illustrates the ways words, numbers and assumptions can get scrambled if left undefined. Ruff will start by giving students the set of numbers 1, 2, 4, 5, 6 and ask them whether there is a gap in it. The most obvious answer is the number 3.

Then she will provide students with a new set of numbers -- 1, 2, 3, 4, 5, 6 -- and ask the same question.

“The word ‘gap’ is not defined,” she said. Ruff then asks students how to define that word in order to show either the existence or the absence of a gap. The numbers also can be defined, perhaps to include quarters and decimals. The point, she said, is to press students to come up with and clearly state definitions and ask students to think about which ones are better than others and why.

"Mathematicians have to be precisely accurate," she said. "It’s so difficult to do without saying something that’s in some way wrong.”

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When course requirements at the Massachusetts Institute of Technology shifted 10 years ago, faculty members in the mathematics department found themselves with a new task in their job description. Not only did they have to teach their students to solve equations; they also had to instruct them in writing and communicating effectively on the subject.

This change in duties -- which mirrored similar shifts in the teaching of discipline-specific writing at other institutions -- gave rise to a host of new challenges, from the administrative to the pedagogic, said Haynes Miller, professor of math at MIT. The math faculty there had to learn how to teach the subject from a different perspective -- one in which words, not just numbers and symbols, are given emphasis.

Arriving at a common definition of effective teaching of writing and communication courses proved to be another obstacle, said Miller, because these classes were taught in seminars, many of which followed unique syllabuses or reflected the preferences and styles of the faculty and students in the courses.

Now, a decade later, members of MIT’s faculty have been developing -- with money from the National Science Foundation -- a website, the Educational Collaboration Space, that is meant to be a forum for those teaching communication skills to mathematics students. Faculty will be able to crowdsource their ideas, post lessons, exercises and classroom examples, reflect on their experiences, and develop some consensus about what works.

“I believe it will be an enormously helpful resource,” said Mia Minnes, SE Warschawski Visiting Assistant Professor in the math department of the University of California at San Diego (she also worked on early iterations of the project at MIT while an instructor there). “Teaching these skills is challenging in any context, and even more so for instructors whose background and training is in mathematics rather than communication.”

MIT hopes the web-based tool that its faculty has devised for teaching writing about math will have a wide application to other disciplines or universities. “It has nothing to do writing itself or mathematics. It could be anything,” said Miller. “Here’s a mechanism for them to form a community, and it’s much more immediate than having department meetings.”

This is far from the first time that someone has thought to develop a platform for crowdsourcing ideas about teaching -- either generally or about math. Members of many a math faculty have used wiki pages for this purpose. The Mathematics Association of America has its own digital library of teaching ideas on the subject (the MMA, which is a co-recipient of the NSF grant with MIT, will eventually host the completed site that will be dedicated to teaching communication skills to math students). The effort at MIT, which is still under development, is intended to be a user-friendly platform for teachers. Still, some expressed skepticism that the diffuse nature of the web would ever lend itself to one site's becoming the authoritative repository for anything, as some behind the MIT effort are hoping will be the case.

Mathematicians' MotivesThe idea that students of math should be able to express their ideas in words is at least decades old, though the goal might more often be proclaimed than fully realized. The underlying motivation for wanting to teach math students to write about their subject also varies among the subset of mathematicians interested in the concept.

Some favor teaching aspiring mathematicians how to write because they feel it facilitates learning. “Clarity of expression is connected with clarity of thought,” said Miller of MIT. “You can’t write something clearly unless you understand it. You can’t understand it clearly until you write it.”

Writing can be a particularly useful tool when math students make the transition from computationally focused courses to more theoretical ones, said John Meier, professor of mathematics at Lafayette College, and a co-author (with Thomas Rishel) of

Writing in the Teaching and Learning of Mathematics. "The use of writing assignments in courses in the calculus sequence is a great way for them to begin to see that there are ideas and logical structure underpinning the computations," he said.Others see good mathematical writing as an end in itself. According to this view, mathematicians need to do a better job of writing clearly about the subject when addressing readers -- both within and outside the discipline.

"There’s long been a feeling among mathematicians that we don’t do a good job of communicating to each other and the outside world,” said Stephen B. Maurer, professor of mathematics at Swarthmore College -- whose own thoughts on the subject date to at least 1991, when he wrote an article in

PRIMUS, a journal dedicated to the teaching of mathematics to undergraduates. "Mathematicians have a hard time reading other mathematicians. Everything you can do to make it clearer is valuable."Minnes of San Diego said this lack of clarity can have serious consequences. “Mathematics has proven time and again to be the key to many important achievements in science and engineering,” she said. “However, these achievements typically require collaboration of many people over many years to mature; such collaboration depends on the articulation of technical ideas among the people involved.”

While every discipline has its own codes of language, styles and conventions, math allows writers to explicitly define terms (as in the phrase “let x=4”). These and other facets make the characteristics of good mathematical writing different from generally accepted ideas about good writing on a general subject, said Maurer.

For example, he said, “elegant variation,” or the use of synonyms to add variety and shades of meaning, is frowned upon in mathematical writing. “That is exactly the opposite of what you should do in mathematics because we split hairs over meanings; if you use a different word, a mathematical reader assumes you meant something different and the difference is important," said Maurer. "Writing in general is difficult to do well, but in math it’s especially difficult."

Finding an instructor with the right blend of skills to teach mathematical writing also can be a challenge. While it is one thing to teach writing on a general subject, it is quite another to write about mathematics, said Susan Ruff, lecturer in the Writing Across the Curriculum office at MIT and, some say, the repository of much of the institution’s knowledge in teaching math students how to write about the subject.

Ruff described an assignment that she uses in her classes, which illustrates the ways words, numbers and assumptions can get scrambled if left undefined. Ruff will start by giving students the set of numbers 1, 2, 4, 5, 6 and ask them whether there is a gap in it. The most obvious answer is the number 3.

Then she will provide students with a new set of numbers -- 1, 2, 3, 4, 5, 6 -- and ask the same question.

“The word ‘gap’ is not defined,” she said. Ruff then asks students how to define that word in order to show either the existence or the absence of a gap. The numbers also can be defined, perhaps to include quarters and decimals. The point, she said, is to press students to come up with and clearly state definitions and ask students to think about which ones are better than others and why.

"Mathematicians have to be precisely accurate," she said. "It’s so difficult to do without saying something that’s in some way wrong.”