School of Science and Engineering
TABLE OF CONTENTS
I. WHAT IS A PROJECT REPORT? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
II. WHY DO WE WRITE MATHEMATICAL REPORTS? . . . . . . . . . . . . . . . . . . . . 2
III. HOW DO WE STRUCTURE A MATHEMATICAL REPORT? . . . . . . . . . . . . . 3
IV. SUGGESTIONS FOR WRITING A GOOD PROJECT REPORT . . . . . . . . . . . 4
A. Suggestions for Each Part of the Project Report . . . . . . . . . . . . . . . . . . . . . 4
1. The Title . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2. The Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3. The Main Body or Mathematical Argument . . . . . . . . . . . . . . . . . . 5
4. The Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
B. General Writing Suggestions for Project Reports . . . . . . . . . . . . . . . . . . . . 6
1. Connections and Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2. Important Aspects of Any Mathematical Project Report:
Correctness, Completeness, Spelling, Grammar, Punctuation . . . . 6
3. What is an “Elegant” Mathematical Paper? . . . . . . . . . . . . . . . . . . . 7
4. Acknowledgments for Assistance with Project Reports . . . . . . . . . . 7
V. WORKING EFFECTIVELY IN A GROUP OR TEAM . . . . . . . . . . . . . . . . . . . . 8
VI. APPROPRIATELY USING MATHEMATICAL SOFTWARE
AND GRAPHING CALCULATORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
VII. EVALUATING PROJECT REPORTS IN MATHEMATICS . . . . . . . . . . . . . . . . 10
A. Overview of Evaluating Project Reports in Mathematics . . . . . . . . . . . . . . 10
B. Sample Grading Scale for Written Projects in Mathematics . . . . . . . . . . . 10
The goal of most writing is to communicate something to someone else. In the case of a project report in mathematics, the goal is to communicate some ideas in mathematics, such as a new concept, a new technique, or the solution to a complex problem, to someone who is not familiar with those ideas. The main body or mathematical argument part of a project report must contain appropriate material to support all conclusions drawn about concepts or to support all solutions to problems.
The project or laboratory instructions you are given by your professor may consist of several questions to answer or several steps to follow with your project partners, often while you are in one of the computer labs. However, a project report is not a list of answers or solutions to the problems given. Even though your professor may have listed various parts or numbered questions in the project instructions, this usually does not mean that you should have these parts or questions numbered in your report. Your job is to tie all of these individual parts into a single, cohesive whole. Instead of focusing on the different parts, try to find the overall purpose that your professor had in mind when writing the project questions or instructions. Your written report should communicate this overall purpose to a reader who has about the same level of mathematical experience that you do. You can imagine your reader as a student in another section of the same course, but a student who is not familiar with the problem addressed in your project.
In some of the projects, the questions that you are asked to answer will be leading you through the solution of a single complex problem. In that case, your project report will state and motivate the main problem and describe the solution in detail. In other projects, the questions that you are asked to answer may involve different examples intended to lead you toward an understanding of some general concept or method. In that case, your project report will be a mathematical exposition (somewhat like a textbook exposition) describing the mathematical concept in general terms and illustrating with specific examples. In either case, your report will read as a single document and will be primarily text, with equations and graphics incorporated into the text when appropriate.
II. WHY DO WE WRITE MATHEMATICAL REPORTS?
To Explain to Others
For most of your life so far, the writing you have done in mathematics classes has been primarily as homework or tests, and you have been explaining your work to people who know more mathematics than you do, usually to your teachers. At this point in your education, you know far more mathematics than the average person has ever learned; in fact, you know more mathematics than most college graduates remember. With each additional mathematics course you take, you further distance yourself from the average person on the street. You may feel that the mathematics you do is simple and obvious. (Doesn’t everyone know what a function is?) However, other people who need to understand these concepts may find this same mathematics complex and confusing. It becomes increasingly important, therefore, that you learn to explain what you are doing to others who might be interested: your parents, your supervisors at work, the media.
Regardless of perception, the mathematical sciences and writing are not at all far-removed from one another. Professional mathematicians, scientists, and engineers spend most of their time writing: communicating with colleagues, submitting proposals, applying for grants, publishing papers, writing memos. Writing well is extremely important to mathematicians and scientists, since poor writers have difficulty writing papers which will be accepted for publication, difficulty communicating effectively with employers, and difficulty obtaining funding for proposals. Writing well is extremely important to engineers, since poor writers have a difficult time convincing a potential client to hire them and difficulty explaining a product or design to employers.
One of the simplest reasons for writing in a mathematics class is that writing helps you to learn mathematics better. By explaining a difficult concept to other people, you clarify your own understanding of the ideas and applications.
An appropriate title demonstrates the purpose and content of the project report.
The introduction gives a basic restatement of the problem, explains the significance or importance of the problem, and ends with a statement of the solution, when appropriate. The introduction of the project report contains what is often called the thesis statement in writing in the humanities and social sciences.
The project report continues with a set-up of the problem and a statement of any basic assumptions. The report outlines the approach to be taken and discusses any limiting conditions that were required to solve the problem. The supporting argument also outlines the reasoning behind the formulas and theorems used and includes explanation of the calculations performed. This main body of the report incorporates supporting graphs, tables, and diagrams in appropriate locations within the text.
The conclusion of the project report critically and carefully analyzes the mathematical solution or solutions to determine appropriateness to the real world setting in which the problem was posed. The conclusion should eliminate any inappropriate solutions with a brief justification of why the solutions were discarded. The conclusion then brings the appropriate solutions back into the real world setting in which they were posed and interprets their significance in that setting.
This portion of the writing guidelines offers more specific suggestions for preparing your mathematics project report. Part A gives some ideas for individual sections of the report.
Part B gives general writing suggestions for mathematical writing and notes some crucial final steps to take before you or your group or team submits the final project report.
While you are preparing any project report, the textbook for your mathematics course will serve as a good example of appropriate mathematical writing. Study some of the examples in your text to see good use of explanation, connections, transitions, centering of formulas, and other writing and formatting techniques.
1. The Title
The title of your project report should be appropriate in tone and structure to a formal mathematical presentation, such as a section of your textbook. It should have some descriptive information about the problem to be solved and the general concepts you applied in solving the problem. The title should not be too wordy, but should give the reader enough information to anticipate the type of problem and the approach you used in solving the problem.
2. The Introduction
In the introduction of your project report, you will first explain what the problem is, and you will often try to convince the reader that the question or problem is an interesting or worthwhile problem to solve. The introduction will often include the result, answer, or solution to the problem, even though the solution process itself will not be explained until the main body or mathematical argument section of your report. For some projects, you may choose to leave the reader in suspense and not give the solution until later in the report.
Clearly restate the problem to be solved. Do not assume that the reader knows the background of the problem or question. You do not need to restate every detail, but you should explain enough so that someone who has never seen the assignment can read your paper and understand your work, without any further explanation from you. Outline the problem carefully.
Give some motivation to the problem. Try to answer the following questions: Why would a person outside your mathematics class be interested in knowing the solution to this question? Where does this issue arise in the real world? In other words, give the reader some context for the project.
When possible, state the result in a complete sentence which stands on it own. In most project reports, the answer or solution will be in the introduction. If you can avoid variables in your statement of the solution, do so; otherwise, remind the reader what the variables represent. If your solution does not appear until the end of the paper and you have made any significant assumptions in the solution, restate them at that time. Do not assume that the reader has actually read every word and remembered all of the details. Also, if you do see the “bigger picture” or think that the project has led to your understanding of a major concept (the “Ah-Ha!” experience), the introduction would be a good place to discuss that “bigger picture” for the reader’s benefit.
3. The Main Body or Mathematical Argument
In a mathematical argument, we want to persuade the reader that our solution to the problem or analysis of the mathematical property is correct. We often use calculations, statements of previous results, and accompanying visuals (such as graphs, diagrams, and tables) to help us accomplish this.
Provide a paragraph which explains how the problem will be approached. Carefully outline the steps you are going to take, giving some explanation of why you are taking that approach. It is nice to refer to this paragraph later when you are well into your calculations, to help the reader follow the sequence of steps you are taking.
Clearly state the assumptions which underlie the formulas that you are using. For example, what physical assumptions do you need to make? (No friction, no air resistance? That something is lying on its side, or far away from everything else?) Sometimes things are so straightforward that there are no assumptions, but not often.
Clearly label diagrams, tables, graphs, or other visual representations of the mathematics, if these are used. In mathematics, even more than in literature, a picture can be worth a thousand words, especially if it is well-labeled. (Please see section VI on appropriate use of software in creating visuals.) There are certain techniques you should use when adding visual components to mathematical writing. For example, label all axes in graphs, usually with words. Give a diagram or graph a title describing what it represents. It should be clear from the picture what any variables in the diagram represent. The idea is to make everything as clear and self-explanatory as possible. Incorporate your graphs, tables, or pictures into the body of your paper. Remember to introduce the visual aid within the writing and to provide some context as to why it is important in the discussion.
Define all variables used. The more specific you are, the better. State the units of measurement. Clarify words such as “position” (Do you mean height above the ground?) and “time” (Do you mean time since the experiment started?).
Explain how each formula is derived or give a reference in which the formula can be found. Some formulas, such as the formula for the area of a circular region, are very well-known and just need to be identified. Others should be derived as part of your mathematical argument, or a source provided. Long formulas are often easier to read if they are centered on a separate line of your text. (See your textbook for good examples of formatting text involving long formulas.)
4. The Conclusion
This is the part of your project report in which you will interpret and explain the mathematical solution to the problem or question presented in the original assignment.
Watch for solutions that do not make sense in the “real world.” When you eliminate solutions which would produce negative time or negative length, for example, explain your reasoning.
Watch for mathematical solutions that need interpretation to make sense. Negative current, for example, would need explanation based on the set-up of the original problem. In your interpretation, always explain any changes you are making to the calculated solutions.
Throughout the project report, it is important to develop a clear exposition that enables the reader to progress as smoothly as possible from the introduction through the mathematical argument to the conclusion. In mathematical writing, it is particularly helpful to clarify connections and transitions with words and phrases such as the following:
Š Therefore (also: so, hence, accordingly, thus, it follows that, we see that, from this we get, then)
Š I am assuming that (also: assuming, where, M stands for; in more formal mathematics: let, given, M represents)
Š Show (also: demonstrate, prove, explain why, find)
Š This formula can be found on page ____ of ________.
Š If (also: whenever, provided that, when)
Š Notice that (also: note that, notice, recall)
Š Since (also: because)
Throughout the work on the project and especially at the stage of the final project report to be submitted, each member of the group or team is always responsible for carefully checking the following major aspects of the paper.
Š Be certain that the mathematics is correct.
Š Be certain that you solved the original problem given or completely answered the fundamental question in the assignment.
Š Be certain that the spelling, grammar, and punctuation are correct.
It may surprise you that many students lose points on projects as a result of spelling, grammar, and punctuation errors. These detract considerably from the quality of the project. Be sure to spell-check and proofread your work. In addition, ask a friend to read the project report for content as well as error detection.
Mathematical formulas are clauses or sentences and need proper punctuation. Put a period at the end of a computation if the computation ends the sentence. Use a comma if the formula or computation is a clause or other part of a sentence. (You will find many examples of such uses within your textbook. Try to follow the format demonstrated in the text.) Do not confuse mathematical symbols with English words. The symbols = and # are common examples of such misuse. The symbol “=” is used only in a mathematical formula or equation. Otherwise, the word equal is written out.
3. Finally, always keep in mind when writing any mathematical paper that “elegant” mathematics papers are the ones that are the easiest to read: clear explanations, uncluttered expositions on the page, well-organized presentation.
4. Give Acknowledgment to Those Who Have Assisted With Your Project Report
It is extremely important to acknowledge the help provided by others, as well as citing appropriate references. In your project report, you should acknowledge any of the following resources: any book you used, any computational or graphical software which helped you understand or solve the problem, any student with whom you spoke (whether in your class or not), any assistance provided by the Writing Center, any proofreaders, and any professor with whom you spoke. Be as specific as possible.
For example, the Mathematics Department faculty thank the following people for their inspiration and assistance in our preparation of these writing guidelines for mathematics projects:
Dr. Annalisa Crannell of Franklin & Marshall College, who graciously allowed us the use of her “Guide to Writing in Mathematics Classes” as the basis for our own writing guidelines development.
Dr. John Bean, Professor of English and Consulting Professor of Academic Writing at Seattle University, for his generous time commitment and thoughtful work with the Mathematics Department in developing both these guidelines and the “Grading Scale for Written Projects in Mathematics.”
The Seattle University Science and Engineering Project Center and the Writing Center for use of their Writing Guidelines, particularly the suggestions for writing as a team.
V. WORKING EFFECTIVELY IN A GROUP OR TEAM
Most of your mathematics projects will be carried out as a group or team. Everyone in the group should be actively engaged in work on the project, both during computer laboratory hours and outside the lab. The ability to work successfully in a group is a crucial skill for people in almost every career. The projects in mathematics provide an opportunity to develop your teamwork skills, while project report writing should increase your understanding of the concepts studied and your ability to apply these concepts to problem solving.
The Science and Engineering Project Center and the Writing Center, in their Writing Guidelines for senior design teams, offer the following suggestions for working as a project team. (These have been adapted slightly to the writing of project reports in mathematics.)
1. Do as a team what teams do best. Do as individuals what individuals do best.
Teams are good at the following:
Š Brainstorming for ideas
Š Achieving creative solutions to problems
Š Planning and organizing activities and setting goals
Š Developing an organizational plan for a document and determining what ideas go into which parts
Š Serving as readers who give feedback on drafts
Š Seeing how individual pieces fit into the whole and making recommendations for improvements of individual pieces
Teams are not good at the following:
Š Writing drafts of individual sections
Š Editing for uniform voice and style
2. Recognize that the pieces of any document make sense only in relation to the whole. Never draft individual pieces until all team members understand the whole.
3. Talk about your document before you write it. As a team, talk your way through each
section of the document. Working together, take notes and plan each section of the document. Articulate its purpose and its content. Each member of the team should feel qualified to write any section of the document. The Writing Center can facilitate this crucial organizational step for project reports.
4. If possible, all team members should use the same word-processing program and follow the same plan for font sizes, heading styles, and formatting. This will make it easier to combine sections of the report.
5. Assign individual team members the responsibility of drafting sections of the document.
Copies of the drafts should be made for all group members. At a team meeting, members should review drafts for accuracy, completeness, clarity, development, and style. Individual writers should then revise their sections. One team member who writes well should do final editing of the document so that it seems written in one voice.
6. Write your project report as you proceed with your work, rather than putting off writing
until the very end. The act of trying to write a section of the document can clarify your thinking and suggest additional paths to follow. The more you write as you go along, the easier it will be at the end to produce an excellent project report.
VI. APPROPRIATELY USING MATHEMATICAL SOFTWARE AND
Every mathematics course makes use of graphing calculators, mathematical software, or both. These can be particularly helpful in working on mathematics projects. In many courses you will have regular computer laboratory days, using mathematical software such as Mathematica, Joy of Mathematica, MATLAB, or Minitab. You are expected to develop some level of proficiency with the provided software, although you are not expected to be an expert in software use. However, you should develop the competency needed to use the software or your graphing calculator to help solve problems. In some classes, you will be expected to learn to use software to present your papers in a neat, professional manner. In Mathematica, for example, you can create entire project reports, if you wish. Alternatively, you can cut and paste into a word processing document, created with software such as Microsoft Word. Sometimes, you may want to sketch a picture to illustrate your writing in a way that the laboratory software does not do, or does not do well. In that case, a program such as Mac Draw or Paint can be useful.
It is important to remember that projects are not simply an exercise in software use. A project report should never be just a sequence of graphs, tables, or other computer output, without appropriate written discussion of the concepts involved.
Include visuals such as figures, graphs, charts, or tables only if you or your team can explain why a reader needs them. Visuals for their own sake confuse readers rather than help them. The document should be able to stand on its own with all visuals removed, because essential information displayed on the visuals should also be discussed verbally in the document itself.
Please see earlier sections of these Writing Guidelines for suggestions on including graphs, tables, diagrams, and charts in your project reports. Your course textbook is also a good source of examples of visuals included appropriately within text.
VII. EVALUATING PROJECT REPORTS IN MATHEMATICS
A. A Brief Overview of Evaluating Project Reports in Mathematics Courses
In the Mathematics Department at Seattle University, most courses from freshman Core classes to senior-level courses for mathematics majors have labs or projects, often carried out in groups or teams, with well-written project reports required. These project reports are evaluated in a variety of ways, based on the level of the course and the nature of the particular project. Each course instructor will give students information about the evaluation of projects in that class, as well as specific expectations for the project reports. However, grading of any project report will involve the structure of the argument, the quality of the argument, and the clarity and professional appearance of the presentation.
Goals for students in all mathematics courses include development of the ability to communicate mathematical concepts, both orally and in writing. Group work on projects and the writing of project reports are crucial parts of this development process. Therefore, the evaluation of projects is based on communication of ideas, as well as correctness of the mathematics.
B. Grading Scale for Written Projects in Mathematics
Dr. John Bean, Professor of English and Consulting Professor of Academic Writing at Seattle University, worked with the entire Mathematics Department faculty to develop the Grading Scale for Written Projects in Mathematics which appears on the following page. This grading scale describes in the left-hand column the attributes mathematics faculty members consider appropriate for an excellent project report. As you read from left to right in each row, reasons are given for higher to lower scores in each aspect of the report.
While not every project will be designed to fit this grading model, studying the grading scale will help you apply the writing guidelines in this booklet and will help you understand other grading models provided by your instructor for specific projects in a particular mathematics class.