| | | The big picture |
A well-organized essay should have an introduction that states the thesis, a middle section that develops the argument, and a concluding section that recapitulates the theme. "First you say what you're going to say, then you say it, and then you say what you said."
Technical research papers often do not follow this model, but they probably should. It is useful to start a research paper with an introductory section that summarizes the results. A good way to close a research paper is to discuss open problems and directions for future investigations.
It is often said, with justification, that you should write from experience. Give concrete examples to support your points. Instead of writing "mathematics is a foundation for many careers", write "studying mathematics can lead to careers in accounting, economics, chemistry, ...". Better yet, write "Arthur Pancoe, one of the nation's ten best stockbrokers, attributes his success to the analytical skills he learned as a mathematics student."
Know your audience. You should write differently for high-school sophomores, for college mathematics majors, and for graduate students. Are your intended readers conversant with the concept of prosthaphaeresis, or are they uncertain about the notion of radian measure? You should adjust your exposition to fit the knowledge, background, and sophistication of your expected audience.
If we are ready to tolerate everything as understood, there
is nothing left to explain; while if we sourly refuse to
take anything, even tentatively, as clear, no explanation
can be given. What intrigues us as a problem, and what will
satisfy us as a solution, will depend upon the line we draw
between what is already clear and what needs to be
clarified.
--Nelson Goodman,
Fact, Fiction, and Forecast
After you finish a writing task, put your work aside for a while. When you read your paper again later, you will find ways to improve the writing. Very likely you will find places where you did not write what you meant or mean what you wrote. For example, the following famous sentence, intended for a review in Mathematical Reviews, never saw print because the author realized that it did not say what was meant.
This paper fills a much-needed gap in the literature.
The Math 696 course
pages were last modified April 5, 2005. | | | The big picture |