MATH 490

In the first part of the semester, you will write reviews of four mathematical articles. A review (sometimes called a critical review) of a scholarly article is a paper that summarizes, analyzes and evaluates the article. Despite what the word may suggest, a critical review is not (usually) a criticism of an article, but rather report in which you summarize an article, explain its purpose, express your reaction to it, and evaluate how well the article meets its goals.

In the beginning, you may have three main questions: "How do I find an article?" and "How do I write a review of it?" and "How will it be graded?" This page is intended to answer those questions.
Finding an Article
Everyone will read the same article for the first review. It is Six Ways to Sum a Series, by Dan Kalman. It was published in the College Mathematics Journal in 1993, and won the 1994 George Pólya Award.

For the remaining reviews, you will choose your own paper to read. Below are links to lists of articles that have been awarded prizes by the Mathematical Association of America (MAA). You may write a review of any article on these lists. (You may also report on an article not listed here, but if you wish to do this, please check with me first.) Browse these lists and look for titles that suggest the article may be of interest to you. Then download it. You may want to download several interesting articles in case your first choice does not work out. The library may have copies of the articles for which no electronic full-text is available.
This award is given to the author of an outstanding expository article on a mathematical topic by a member of the MAA.
Established in 1964, this award recognizes authors of articles of expository excellence published in The American Mathematical Monthly or Mathematics Magazine.
Established in 1976, this award is given to articles of expository excellence published in Mathematics Magazine.
Established in 1976, this award is given to articles of expository excellence published in the College Mathematics Journal.

You can download full text versions of many of these articles from the Mathematical Association of America's Digital Library. Here is a link to their Writing Awards page. If this page does not have a full-text version of the article you are looking for, try searching for it on Jstor, a very large database of journal articles.

In choosing an article, you may find it helpful (at least at first) to avoid articles that are too elementary or too advanced. If your article is too elementary or lacks significant mathematical content, you will probably discover that you cannot find enough to write about. At the other extreme, if your article is too advanced you may be faced with the problem of writing about something you didn't understand. Aim for the middle ground. The College Mathematics Journal or Mathematics Magazine usually feature accessible articles with significant content. Articles in Math Horizons are sometimes too watered down. Those in American Mathematical Monthly can be challenging if you haven't studied the requisite background, but they are often quite accessible.

Writing a Review
Once you have selected an article that looks interesting, you should follow the following guidelines.
READ. Obviously, reading your article is the first step in writing a review. But it is not always quite so obvious how to read it. Here are some suggestions.
Start early. Full understanding of a mathematics article can take time. You may need several days to a week to fully digest its content.
Read actively. Think hard about what you are reading. Read for understanding. Have pencil and paper by your side as you read, and check details as you go along. Do lots of scratch work. If a result or theorem is presented, create your own examples that the theorem applies to. Such examples improve your understanding and can be incorporated into your review.
Reread. In mathematics, you often must reread a certain passage numerous times before you begin to understand it. If you are really stuck, it can be helpful to read ahead to get a feel for where the article is going, but you should then return to the part that was giving you trouble.
Consult references if necessary. For example, if the article talks about "positive-definite matrices," you may need to look this up in a linear algebra textbook.

ANALYZE. After reading, ask yourself the following types of questions.
What is the article's main point?
What kinds of mathematical knowledge is required of the reader?
How are the mathematical results achieved? Is there an especially clever or ingenious line of reasoning?
Does the article merely summarize existing ideas and results? Does it offer a new interpretation of existing knowledge, or does it set forth totally new ideas?

EVALUATE. Formulate your responses to the following types of questions.
Is the article clearly written and well-organized?
Were important terms clearly defined?
Did the author present an appropriate amount of background information?
Did the article help you understand its topic? Did it make you want to learn more?
Did you find any flaws or omissions in the author's reasoning?
Are there other points of view that the author did not address?
Did you fully understand all of the article?

WRITE. Now you are ready to begin writing. There are no definite rules here, but your paper should meet all the requirements stated below.
Begin with the full bibliographic citation (author, title of journal article, name of journal, volume, issue, date of publication, pages).
Summarize the article. (Explain what it accomplished.)
Your review should contain some mathematical content. If you find a particular proof or line of reasoning especially appealing, by all means include it in your review. Explain it in your own words. Feel free to flesh out details -- or summarize -- as appropriate.
One of the main reasons for having you write a review is to deveolp your skills at explainig mathematical ideas. You should assume that I don't know anything about the topic, and that the purpose of your review is to explain it to me.
Address questions of the type outlined in headings 2 and 3 above.
Aim for three to four pages (not including cover sheet) double-spaced with normal font (about 11 point) and margins (about 1 inch).
Every sentence should convey some clear meaning. Do not use vague ``filler" sentences.
Use the active voice.
Avoid unnecessary quotations. Convey ideas in your own words.

POLISH. Once you are done with the main task of writing, you need to put the finishing touches on your paper. Please consider the following points.
Important: Proofread your paper carefully. Look for typographical and grammar mistakes. Think about your sentence structure and make any necessary improvements. It's a good idea put the paper down for a day or two before you do this. If you give yourself time, you will think of improvements almost by accident.
Use appropriate mathematical typesetting. Writing x2 as x^2 is very, very bad.
Insert references as needed.
Create a cover sheet with your name, title, the class and date.
Important: Attach a copy of the paper your critique is based on. I'll want to cross-check your critique with the article. (But you don't have to do this for the first review, since I already have a copy of it.)
Important: Please staple or otherwise bind everything together before you submit it. Presentation can make a difference.

The following table indicates how your paper will be graded

} 40 points
Mathematical content and accuracy
} 40 points
Grammar, punctuation, sentence structure
5 points
Typesetting (x2 instead of x^2, etc.)
5 points
Inclusion of the article you are critiquing
5 points
Title page, binding
5 points
It is due at the beginning of class on the appointed day. A paper submitted later than that automatically loses 10 points. An additional 10 points is deducted for every day it is late. Further, I may spend much less time reading a late paper, and consequently I may offer little or no feedback.