• "Modular elliptic curves and Fermat's Last Theorem", Andrew Wiles, Annals of Mathematics, vol. 141, 1995, pp. 443-551.
• "Sharp Holder estimates for the solution of dbar on ellipsoids and their complements via order of contact", Julian F. Fleron, Proceedings of the American Mathematical Society, vol. 124, no. 10, October 1996, pp. 3193-3202.

• Message - An expository or survey article is an article that gives a fairly broad overview of a specific field, topic or problem. It usually highlights the motivation and development, the major breakthroughs, the critical results, the major open questions, and the connections and/or applications of the field, topic or problem in question. While often fairly specific, the goal of an expository or survey article is to give the audience an appreciation of these issues without undertaking a detailed study of a textbook, lecture notes, and/or an entire collection of research papers. Generally expository or survey articles are 5 to 30 pages in length.
  Students are often asked to write (perhaps more limited) expository or survey articles on a major topic they are studying as a way of helping them attain a broader, more comprehensive overview of this topic.

• Writer - A professional mathematician authoring an expository or survey article is faced with the daunting task of weaving the development, motivation, major breakthroughs, critical results, major open questions, and connections of field, topic or problem into a coherent whole. This is a difficult enough task if one is writing an extensive book. Yet the author of an expository or survey article is forced to do this within the confines of a fairly short article that must be accessible to a broad, non-expert audience. Hugo Rossi, then editor of the Notices of the American Mathematical Society noted this difficulty when introducing expository or survey articles into that journal in 1995, saying:
  "It is extremely hard for mathematicians to do expository writing. It is not in our nature. In fact, the very nature of mathematical meaning and grammar militates against it. However, this puts us at a distinct disadvantage relative to other sciences... Good exposition should be valued, not only for the success in communication but also as evidence of real mathematical insight. It is no accident that among our greatest mathematicians are our greatest teachers and expositors." (Notices of the American Mathematical Society, vol. 42, no. 1, January, 1995).
  Student authors of expository or survey articles should be aware of those same issues that face the professional mathematician but should not be overwhelmed by them. Students are often assigned survey or expository topics to help them articulate a unified or holistic view of a given area of study. This is meant as an educational task, the finished product is not expected to have the same depth, insight, or sophistication that a professional mathematician might bring to the task.

• Audience - Expository or survey articles are directed to broad, non-specialized, non-expert audiences. The experts in a given area have little direct use for expository or survey articles -- they already know and understand these broad themes in depth. However, it is often of great importance for these experts to share the results, questions, methods and breakthroughs in their area with others, and expository or survey articles are how they do this.
  There remains a tremendous range in the level of mathematical sophistication assumed in expository or survey articles. Books like What is Mathematics? (see below for details), which are written to help laypeople develop an appreciation for mathematics, often are an entire collection of survey articles united under a common theme. Papers like "An Introduction to Fractals" (again, see below) are intended to provide a broad cross-section of mathematicians, engineers, physical and biological scientists, and others a substantive introduction to a new, but rapidly emerging field. Still others, like those that have been included in each issue of the Notices of the American Mathematical Society since January of 1995 (again, see below) are directed to professional mathematicians. Although they are expository in the sense described here they remain at a level that only those with significant graduate training in mathematics can understand.

• Examples - What is Mathematics? by R. Courant, H. Robbins, and I. Stewart, (other details); "An Introduction to Fractals" by Jenny Harrison (from Chaos and Fractals: The Mathematics Behind the Computer Graphics edited by R.L. Devaney and L. Keen, American Mathematics Society, 1989), "An Update on the Four-Color Theorem by Robin Thomas, Notices of the American Mathematical Society, vol. 45, no. 7, August 1998; "Fractal Image Compression" by Michael Barnsley, Notices of the American Mathematical Society, vol. 43, no. 6, June 1996.

• Journals containing other examples - Notices of the American Mathematical Society, Mathematical Intelligencer.

• Message - As noted in the introduction to this Writing in Mathematics section, over 70,000 research papers, books, monographs and survey articles are published in mathematics and the mathematical sciences each year. Without systematic reviews summarizing and critiquing this work and the critical classification system of the "Mathematics Subject Classification System" of Mathematical Reviews and Zentralblatt für Mathematik, this volume of written work would cripple mathematics. (This dilemma even has a name -- Ulam's dilemma. See pp. 20-3 of The Mathematical Experience by Davis and Hirsh for details.)
  Reviewing the significant research literature in mathematics is a remarkable task undertaken by the American Mathematical Society and published in the journal Mathematical Reviews. As noted above, Mathematical Reviews publishes some 70,000 reviews each year. These reviews are generally brief summaries of the main results of the paper under review and how these results fit into the existing literature.
  In addition to the brief reviews in Mathematical Reviews, almost every journal in mathematics publishes reviews in several categories, including reviews of: research articles, books, monographs, expository articles, lecture notes, and computer software for mathematical research or teaching. Generally these reviews serve both to summarize or outline the content of the reviewed work and to provide a critique of this work. These reviews range in length from a few paragraphs to a dozen pages.

• Writer - Almost universally the writer is from the same area of mathematics that the author of work being reviewed is. That is, research mathematicians in real analysis (26xx) would write the reviews of research papers with the subject classification 26A30 - Cantor functions for Mathematical Reviews, mathematics educators review books on mathematics education for Journal for Research in Mathematics Education, etc. Notable mathematicians in a given field are generally asked to review survey or expository books and papers in that field.

• Audience - The audience of a review is generally the same as the writer of the review as well as the author of the book or paper being reviewed. Only research mathematicians read Mathematics Reviews for example. The audience for reviews of survey or expository books and papers is generally broader. For example, books like Innumeracy: Mathematical Illiteracy and It's Consequences by John Allen Paulos are reviewed in the New York Times Book Reviews and other mainstream sources.

• Citations of Examples:
  • "Book Review: The Bible Code" by Allyn Jackson, Notices of the American Mathematical Society, vol. 44, no. 8, September 1997.
  • "Book Review: The Mystery of Numbers," by Julian Fleron, MAA OnLine, July 1998.

• Journals Containing Further Examples - Mathematical Reviews, American Mathematical Monthly, Mathematics Magazine, College Mathematics Journal, Notices of the American Mathematical Society, Bulletin of the American Mathematical Society.

• Message - Because experiential, experimental observations provide the dominant framework within which science progresses, lab reports that document these laboratory experiences are critical to science courses. [Links to science pages hereabouts.] While observations, examples, empirical data, inductive reasoning and inferential reasoning play an important role in the development of mathematical ideas, conjectures and theories, deductive reasoning utilizing formal proofs is the dominant methodology. Thus laboratory reports are not broadly utilized in the world of research mathematics. Nonetheless, they do serve as an increasingly important vehicle for communication within mathematical classrooms.
  Laboratory reports are becoming widely used in calculus courses, as well as in courses in statistics, mathematical modeling, and several other areas of mathematics. Unlike the sciences, where laboratory serve a fairly uniform purpose, laboratory reports in mathematics serve a broad variety of educational purposes. These purposes include: mathematical modeling, real-world applications, introduction to proofs, and data collection and analysis.

• Writer - The writer of a mathematical laboratory report is almost always a student enrolled in a course where laboratories are used for educational purposes in the learning of mathematics.

• Audience - Because mathematical laboratories are usually used for educational purposes, the audience of a mathematical laboratory report is generally the writers themselves, fellow students, and the teacher of the course.

• Format and Structure - As mentioned in the discussion of message of a laboratory report, laboratory experiences in mathematics serve many different purposes. Accordingly, laboratory reports in mathematics can have many different formats and structures. We discuss several here.
  • Data Collection and Analysis - Laboratory experiences involved in data collection and analysis include topics in statistics, in calculus where one considers approximation of limits or integrals using Riemann sums, and in numerical or complex analysis where one investigates specific trajectories arising via Newton's method. A significant portion of the experience is generally devoted to the actual data collection and the subsequent analysis is generally inductive or inferential. Hence, such laboratory experiences closely resemble the typical science laboratory experience. Consequently, the format and structure of a laboratory report for such an experience should closely resemble the scientific laboratory reports considered in other sections of this Writers' Guide.
  • Introduction to Proofs - Laboratory experiences can be structured to help introduce students to proofs. For example, in Calculus I students might be asked to generalize their use of the definition of the derivative in several specific examples to a general proof of the Sum Rule for Derivatives. In this case the deductive reasoning is critical and the final laboratory report should resemble a research paper in mathematics more than it would a typical scientific laboratory report.
    
  • Real-World Applications - Laboratory experiences may be structured around the application of a mathematical method to a real-world situation.
  • Mathematical Modeling - Mathematical Modeling is a critical area of mathematics where one creates a mathematical model of a real-world situation and then analyzes the mathematical model in the hopes of making appropriate predictions about the behavior of the situation modeled. Critical phases of mathematical modeling are: creating a mathematical model of the real-world situation, analyzing the mathematical model, interpreting the results of the mathematical analysis in the real-world situation. It is critical to note that both the first and last stage involve critical elements of the scientific process: defining the problem, collecting data, making hypotheses, testing hypotheses, etc. As such materials and methods, implications, data, and other topics critical to a scientific laboratory report should be included in a laboratory report on mathematical modeling. However, the phase of mathematical modeling that involves mathematical analysis should generally be deductive. In this stage one should proceed deductively, either by providing explicit details establishing the validity of the needed work or providing precise citations that definitively justify the necessary results. Thus a laboratory report will generally include elements of both a scientific laboratory report and a research paper in mathematics.

• Message - This is a category that we have all had experience with, yet we often forget its basic premise: Look how I have solved this problem. Instead of trying to communicate the solution of a problem as it was attained, ideas mingling with computations, equations providing graphs, intuition yielding strategies, writers typically provide only a sketchy outline based on equations and computations. Such an approach yields incomplete, incoherent, and often mathematically incorrect solutions that are inappropriate.

• Writer - Students of mathematics are the preeminent authors of solutions to problems. The solutions teachers present at the board in a classroom setting generally more like outlines of solutions than what would expect for a written solution to a problem. Teachers and textbook authors often write solution guides for students, and although they differ dramatically in quality, they can give you an idea how to write solutions to problems. Mathematicians, professional and amateur, join the many students who submit solutions to problems that are posed in mathematical journals like The American Mathematical Monthly, although these solutions resemble research papers in miniature more than they resemble a problem solution that a typical student of mathematics would recognize.

• Audience - In order or importance, the audience of written solutions to problems of the type that are encountered in a mathematics classroom is: the writer themself, a teacher, and fellow students. A critical student misconception is that the teacher is the primary audience for all written solutions to problem and that because the teacher "knows what I mean" it is sufficient to simply supply a few equations and/or calculations. In fact, this provides neither a clear, complete, nor appropriate solution to the problem.
  In the setting of a test, the entire point of assessment is for the student to demonstrate their understanding and knowledge. In most cases students' understanding of a mathematical idea, method, or strategy is shrouded by their inability to express themselves coherently and it is inappropriate to assume "the teacher knew what I meant."
  When solutions to problems are part of a typical "homework assignment" the assumption is also that the teacher or teaching assistant is the primary audience. Again, this is resoundingly false. The writer is the primary audience. The teacher or teaching assistant grades homework only i) as a stick to insure that you are keeping up with the work, and ii) to provide you feedback so you can improve your solution of problems on tests and exams. So why would a student write solutions for themselves? There are several critical reasons:
  • As noted throughout this writing guide, writing in mathematics and science is a powerful learning tool. Writing a coherent, complete, and mathematically correct solution to a given problem forces the author to synthesize, or bring together, all aspects of the problem and the proposed solution into a unified whole. This is a critical step in the learning process.
  • Writing solutions to problems documents your work.
  • Your solutions to problems are a critically important tool when solving subsequent problems, when reviewing subject areas, and when studying for tests and exams. The skeletal computations and calculations that, inappropriately, make up most students' solutions are of no use if only they have been dutifully recorded. For they rarely provide enough of a clue how to solve a problem so the student can review the given problem without solving the problem all over again from scratch.

• Format and Structure - In general written solutions to problems should be reflective of the authors successful solution of the problem. They should contain the overall strategy for the solution and all of the relevant details. The completeness and coherence of the solution that existed in your mind and on your scratch paper when you finally became convinced you had correctly solved the problem should be clearly apparent from your written solution. While it is possible to provide too many details, it is generally the case that the students who provide too many details are outnumbered by a factor of 100 to 1 by those students that do not provide sufficient details or a coherent indication of their solution strategy. Until your teacher or fellow students begin to complain, more is better.

• Advice to Writers - Written solutions to problems are not the jumbled mix of formulas, calculations, and dead-ends that you investigate while trying to solve a problem. You should not expect to be able to work out problems on the same paper that you will turn your written solutions in on unless you write in pencil and have a really big eraser. Work on scrap paper while you are trying to solve a given problem, and once you have found a solution synthesize, or bring together, your strategy and all the relevant details into one complete, coherent solution.
  The following checklist should be helpful when writing up solutions to problems:
  • Does my solution solve the question or problem as it was stated?
  • Is my solution written in complete sentences, mathematical sentences as well as English sentences?
  • Does my solution demonstrate a clear understanding of both the problem at hand and the solution being presented?
  • Have I given adequate justification in support of my solution to the problem?
  • Does my solution provide a precise, coherent, and well organized solution that would be intelligible to somebody who might not know the exact details of the problem at hand?
  • Have I checked the mathematical correctness of my solution against the answers in the back of the book, another student's answers, or with the teacher?
  • Are all quantities and referents, including both mathematical quantities and pronouns referring to objects, clearly and unambiguously identified?
  • Is the solution clearly labeled so it is evident what problem it solves?

• Examples - Several examples from a variety of courses are given here.

  • Mathematics for Liberal Arts.

    • From Chapter 1, Lesson 1, Set II of the book Mathematics: A Human Endeavor by Harold Jacobs:
      9, 10. The ball will have the simplest path on table number 4, because on this table the ball travels directly from its initial position into the opposite corner pocket. This should happen on any square table.
    • From Chapter 1, Lesson 3, Set I of Jacobs:
      2. The equation 1+3+5+7+9 = 5x5, given in the previous problem, is a valid equation because 1+3+5+7+9=25 and 5x5=25 as well.
    • From Chapter 1, Lesson 1, Set II of Jacobs:
      3. The number of segments in each of the paths given in the table in problem 2 can be found by dividing the length of the table by the width of the table.
    • From Lesson 1, Set II of the manuscript "The Infinite" by Julian F. Fleron:
      15. Based on problems 12 - 14 in this set, it seems that when a piece of paper is folded in thirds repeatedly n times the resulting paper would be 3^n layers of paper thick.
    • From Lesson 2, Set I of Fleron:
      16. As the larger square in the "Square of Aristotle" undergoes one full rotation the smaller square sweeps out a broken, straight line path which consists of four lines and looks like - - - -. Here each of the lines and each of the gaps has the same length as the side of the smaller square.

  • Calculus I.

    • From Chapter 1, Section 1 of Single Variable Calculus by Deborah Hughes-Hallett, et.al.:
      4. If K is proportional to v squared, the a formula representing their functional relationship is
      K=cv ²
      where c is some constant.
    • From Chapter 1, Section 2 of Hughes-Hallett, et.al.:
      12. First find the intercepts of the equation 3x + 4y = -12:
      Setting x = 0 yields 3(0) + 4y = -12, so 4y = -12 and y = (-12)/4 = -3. So (0,-3) is the x-intercept. Similarly, setting y = 0 yields 3x + 4(0) = -12, so 3x = -12 and x = (-12)/3 = -4. So (-4,0) is the y-intercept.
      To find the distance between the intercepts we use the distance formula:
      d = ((x₂ - x₁)² + (y₂ - y₁)²))^1/2 = ((0 - -4)² + (-3 - 0)²))^1/2 = (16 + 9)^1/2 = 5.

• Sources containing other examples - Student solution guides which provide solutions to some of a textbook's problems are available for many mathematics texts. Check with the bookstore or your teacher for availability. Schaum's Outline Series has an extensive series in mathematics. These books outline a given area of mathematics, provide hundreds of worked out problems, and provide hundreds of supplementary problems. Their mathematics series includes over 30 books, including: basic mathematics, beginning calculus, calculus, differential equations, linear algebra, trigonometry, geometry, matricies, real variables [i.e. real analysis], complex variables [i.e. complex analysis], and probability and statistics.

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