riting for a math class strikes many students, and teachers too, as an odd idea to say the least. However, an increasing number of educators have recognized the importance of written composition, especially in lower division and survey courses, for helping students to master and express mathematical ideas. When a math class consists, as it too often does, of nothing more than a collection of techniques to be learnt by rote and regurgitated on exams, then certainly writing about those techniques is superfluous. But when a mathematics course, as it ought, becomes a journey of discovery of mathematical ideas and the importance of those ideas in our appreciation of the world then writing about mathematics can become a powerful component of the learning process.
An increasing number of educators have begun to incorporate a modest amount of composition into their syllabi, with surprising results. The reason for their success is not difficult to understand. Most instructors will readily admit that they never truly understood the basic mathematical ideas that constitute algebra, trigonometry, and calculus until they had to teach those ideas to others. What this fact illustrates is that until an idea is communicated to others it has not really made itself at home in our own minds. Communicating ideas brings them alive, pulling them off the flat page of our memory and giving them solid shape in the dynamic space of the intellect. When students communicate through written composition about the ideas they are learning, they can achieve the same result for themselves.
A second reason for writing in the mathematics classroom is to bring context and background to students appreciation of their subject. Mathematics is rich with its own history, a history moreover peopled by extraordinary characters and punctuated by the steady advance of powerful ideas. Sadly, most students learn the basic math curriculum right up through college largely in ignorance of and untouched by this history. Little wonder that to many students mathematics remains a dead subject, wholly abstract and removed from their lives and interests.
If the Fundamental Theorem of Calculus is arguably one of the profoundest creations of the human intellect and human civilization, should not students learn of the people and history surrounding its inception? But where will they learn of this, if not in their calculus class? Is there no value or interest in learning that Pascal developed combinatorics so as to win at the gambling table? Or what the response of the Pythagoreans was when the irrationality of the square root of two was demonstrated? Or how Alan Turing perhaps saved England during World War II by cracking the German Enigma code, and that he did so only a few years after describing, for the first time, the mathematical idea of a programmable computer, an event that has transformed all our lives? By providing the opportunity to explore these events and learn about these people, even the occasional small writing assignment can transform a student's outlook on mathematics.
Admittedly, for both the student and the instructor, incorporating writing into the math curriculum presents real challenges, not the least of which is the pressure of time. These issues are examined below, and guidance is provided (for students and instructors) for making the writing experience in the math classroom a successful and rewarding one.
am a math teacher, not an English teacher, we hear you cry. Quite right. You are not trained to teach composition, nor are you probably much inclined that way. Nor should you be. But consider: Should the science teacher teach math? Probably not. Nonetheless, you would quite understand the science teachers expectation that his or her students use math correctly when called upon to do so in the science classroom. Our disciplines do not exist in isolation from one another, and expecting your students to be able to compose standard written English on a mathematical topic is entirely within your purview as a math instructor.
There remains the question of how to implement writing assignments in a math-class setting. The issues to be addressed include the structure of writing assignments, the composition standard required of students, and assessment of written work. We will review these in turn.
Our own efforts at implementing writing in the mathematics classroom is ongoing. If you have (or develop) innovative writing assignments for your math students, we would be pleased to hear about them. Please write to us at firstname.lastname@example.org. The assignments we learn of that would be of interest to other instructors will be added to the list of sample assignments above, with credit given to the contributor and a link provided to the contributor's web pages.
THE STRUCTURE OF WRITING ASSIGNMENTS
The first thing to realize is that you have complete freedom here. Writing assignments can be as short or long, simple or complex, as you wish. However, particularly if giving and grading writing assignments is a new venture for you, we recommend keeping things as simple as possible at the outset. When we first tried it ourselves, we began by assigning one or two page biographical sketches of famous mathematicians as extra credit assignments. This is an excellent way to develop a feel for what it is like to assess such assignments. From this beginning, we have experimented with incorporating writing in the math class more thoroughly, with assessment of writing assignments forming an integral part of student assessment for the course. Starting small and simple helps your students also, as they can then become accustomed to the standard you will require of them. Additionally, the simpler the assignment, the less time must be devoted during class (and by your students outside of class) to getting the assignment organized, understood, and completed. To help you get a feel for how writing assignments may be structured, we present here some assignments that we have used over the last several years, together with examples sent to us by other teachers:
We always place these assignments on our department-provided web pages, where students may access them and print them. This encourages students to use the web for research, and permits us to include links to some of these resources. You are welcome to save these sample assignment web pages to your home directory to serve as a template for your own assignments.
Infinity Research Paper
"Large Numbers" Research Paper
Overpopulation Thesis Paper
Logical Fallacies Paper
THE REQUIRED FORMAT OF WRITING ASSIGNMENTS
Although you may be tempted not to require any consistency of style or format from your students, we strongly advise that you do so, for many reasons. First, the (admittedly bland) uniformity of the required style permits the reader (i.e., the grader) to focus on the content and substance of a paper without being influenced (or distracted) by its formatting. Second, the required format promotes clarity, and makes it easy to find important information, such as the author of the paper, the references used, and so on. Third, it lets students know what is expected, and permits the instructor to apply a uniform grading standard vis-á-vis the purely structural aspects of the paper. In short, keeping in mind that we are not trained to teach composition, setting a minimum standard for the purely formal compositional elements frees us to focus assessment of our students efforts on the content rather than the form of their papers.
Because there is not a universally accepted style for mathematical writing (and to the extent that there is one, it addresses exposition at the research level, which is irrevelent here), we have always implemented the Modern Language Association (MLA) academic standard for writing assignments. This is the standard that is taught in freshman composition courses, and the primary reference text is always available in college bookstores. (For flexibilitys sake, we generally permit students trained in the APA standard to use it instead provided they do so properly.) A concise, detailed exposition of the MLA standard for academic papers is included at the end of this article.
As regards other qualities of their compositions, such as sentence structure, paragraphing, and so forth, we allow students more latitude than a composition teacher would do, because such issues are not central to the purpose of a writing assignment in a math-class setting. We do, however, mark down a paper for failure to follow the prescribed format, sloppy presentation, poor grammar, and rampant spelling errors. This simply reinforces a standard that should be common to all academic work.
It is natural to divide assessment of students written work into two parts form and content. By setting a minimum standard for the form of the paper (as outlined above, and as described below in the For The Student section), assessment of the form of the paper is facilitated and made uniform. How content is assessed will vary greatly depending on the assignment. The guiding principle is to ensure that students know what is expected of them, and that assessment of their work is applied as uniformly as possible. The challenge for the instructor is that some students will be very good at written composition, whereas others will be very poor at it. By focusing on content rather than style, you can help to level the playing field in such a way that all your students will get the greatest benefit possible from the assignment.
hen you are told that you will be writing compositions for your math class, you may be surprised, and even feel somewhat resentful. Isnt math class for doing math? Of course it is. But you should look upon your writing assignments as opportunities. They will give you a chance to talk about the math you are learning, which is an excellent way to enhance your knowledge. Your efforts will go more smoothly and be more successful if you approach these assignments the same way you should approach doing mathematics; methodically, and with a spirit of play and discovery. The information which follows is intended to make writing assignments more manageable for you, and to give you an edge in writing the best papers possible.
When writing a paper, concentrate first on the writing itself, and save formatting and proofing till last but then be thorough. Give yourself time to write and rewrite, without being rushed. In an academic paper, searching for what you want to say can take time, and then saying it clearly can take several attempts. The biggest part of your grade comes from organizing your ideas and presenting them clearly. The perfect grade is apt to go to that paper which is free of glaring spelling and grammar errors, is in the proper format, and presents its material clearly, supporting it well from the sources. Study the information in this page thoroughly, and refer back to it each time you write a paper for your mathematics course.
BASIC REQUIREMENTS FOR YOUR PAPER
Papers should always be typed, double-spaced, using either courier or times roman typeface in black 12 point type. DO NOT use any other typeface, color, or size.
The paper used should be stock white, and a standard size (8 1/2 × 11 in. in North America, A4 elsewhere).
Every page should have a 1 inch (2.5 cm) margin on all sides. This is critical to allow the grader to write comments in the margins.
When references are used, a works cited page must be prepared.
Citations are made in accordance with the MLA (Modern Language Association) standard. If you are specifically trained in the use of APA, then the APA style may be substituted, but only with instructor permission.
Every paper should have a title.
The student authors name should appear, together with the page number, in the top right hand corner of every page (this element only may violate the 1 inch margin rule).
The first page should have the students name, the instructors name, the class, and the date in the top left-hand corner of the page (see illustration below).
Titles of books, titles of web pages, and any non-English text must either be underlined or italicized.
DO NOT fully justify text (left justify only). In other words, the right edge of the text should be ragged.
When a complete paragraph or block of text has a single source, place the citation at the end of the paragraph or block of text.
Quotations longer than three lines of text must be indented.
Use of graphs and other images is encouraged. Images that cannot be printed inline with the text (such as hand-drawn graphs, etc.) should be attached on separate sheets, with titles such as Figure 1, and referenced by title in the text. Note that such extra sheets will ordinarily not count towards length requirements.
Alphabetize the works cited page, and ensure that the references are listed the same way they are cited in the text. (See Format for References below.)
The second and following lines of references are indented on the works cited page, not the first line.
The works cited page may be included on the last page of text if room allows. When this is done, separate the works cited from the text by using a solid line, drawn from the left margin to the center of the page. (I.e., this is drawn between the last line of text and the words works cited.)
BE SURE TO CITE YOUR SOURCES IN THE TEXT.
Forget formatting until you have finished doing your research, compiling your sources, and preparing your draft paper. When the paper is essentially finished, then is the time to carefully format it. Finally, proof the completed paper for spelling (use the spell checker!) and grammar. Sloppiness with respect to proper formatting, correct spelling, and good grammar will cause the paper to be marked down.
ollow these formats for references as closely as you can. The guiding concern here is that the reader of your paper be able to find your source and locate the information you used as conveniently as possible.
Book References should list the author, title, and publication information, as follows:
Author Last Name, First Name. Title of Book (ital-
Note that this is the basic form; there are many variations for anthologies, multi-volume works, etc. See the MLA Handbook for Writers of Research Papers for guidance.
icized). City of Publication: Publisher,
year of publication.
Lobdel, Jared. England and Always: Tolkein's World of
the Rings. Grand Rapids: Eerdmans, 1981.
WEB PAGES (& misc. electronic sources)
To cite files available for viewing/downloading via the World Wide Web, give the author's name (if known), the full title of the work in quotation marks, the title of the complete work (if applicable) in italics, the full http, ftp, or internet address, and the date of the visit.
Author Last Name, First Name. "Title of Work." Title of
If the name of the author is not available, the reference may be listed (and alphabetized) by title or subject. Be sure the reference is cited in your text the same way it is listed in the works cited page.
Complete Work (italicized). [protocol and address]
[path] (date of message or visit)
Burka, Lauren P. "A Hypertext History of Multi-
User Dimensions." MUD History."
(12 Nov. 2001)
Bruckman, Amy. "How To Manage Deviant Behavior
in Virtual Communities." ftp://medea.mit.edu
/pub/asb/papers/defiance-chi97 (18 Mar. 2001)
"Writing for a Math Class." Platonic Realms.
(4 May 2001)
Articles from periodicals should be cited using the author's name, the title of the article in quotation marks, the name of the periodical (italicized or underlined), the series number or name, the volume number (if applicable), the date of publication, and the page numbers:
Booth, Wayne C. "Kenneth Burke's Way of Knowing."
Critical Inquiry 1 (1995): 1-22.
s for any class, the papers you write for your math class will invariably reflect the time and effort you put into them. Although we are all familiar with the somewhat romanticized image of the disorganized student burning the midnight oil to finish a paper only a few hours before it is due, it should be remembered that such students are rarely successful. Start working on your paper early, give yourself time to come back to it, and aim to have your draft completed at least a day or two early. This way, you can pretty it up at your leisure. If writing well is a particular challenge for you, we strongly recommend you get a copy of Strunk & White's The Elements of Style. This brief book, easily absorbed in an hour or two over a cup of something warm, will change how you write forever, and very much for the better.
We will leave you with some excellent advice from William Safire:
FUMBLERULES OF GRAMMAR:
Don't use no double negatives.
Proofread carefully to see if you any words out.
Take the bull by the hand and avoid mixed metaphors.
If I've told you once, I've told you a thousand times, resist hyperbole.
"Avoid over use of 'quotation "marks."'"
Avoid commas, that are not necessary.
If you reread your work, you will find on rereading that a great deal of repetition can be avoided by rereading and editing.
Avoid clichés like the plague.
Never use a long word when a diminutive one will do.
Avoid colloquial stuff.