## How to Effectively Write a Mathematics Research Paper

Mathematics research papers are different from standard academic research papers in important ways, but not so different that they require an entirely separate set of guidelines. Mathematical papers rely heavily on logic and a specific type of language, including symbols and regimented notation. There are two basic structures of mathematical research papers: formal and informal exposition .

## Structure and Style

Formal Exposition

The author must start with an outline that develops the logical structure of the paper. Each hypothesis and deduction should flow in an orderly and linear fashion using formal definitions and notation. The author should not repeat a proof or substitute words or phrases that differ from the definitions already established within the paper. The theorem-proof format, definitions, and logic fall under this style.

Informal Exposition

Informal exposition complements the formal exposition by providing the reasoning behind the theorems and proofs. Figures, proofs, equations, and mathematical sentences do not necessarily speak for themselves within a mathematics research paper . Authors will need to demonstrate why their hypotheses and deductions are valid and how they came to prove this. Analogies and examples fall under this style.

## Conventions of Mathematics

Clarity is essential for writing an effective mathematics research paper. This means adhering to strong rules of logic, clear definitions, theorems and equations that are physically set apart from the surrounding text, and using math symbols and notation following the conventions of mathematical language. Each area incorporates detailed guidelines to assist the authors.

Related: Do you have questions on language, grammar, or manuscript drafting? Get personalized answers on the FREE Q&A Forum!

Logic is the framework upon which every good mathematics research paper is built. Each theorem or equation must flow logically.

Definitions

In order for the reader to understand the author’s work, definitions for terms and notations used throughout the paper must be set at the beginning of the paper. It is more effective to include this within the Introduction section of the paper rather than having a stand-alone section of definitions.

Theorems and Equations

Theorems and equations should be physically separated from the surrounding text. They will be used as reference points throughout, so they should have a well-defined beginning and end.

Math Symbols and Notations

Math symbols and notations are standardized within the mathematics literature. Deviation from these standards will cause confusion amongst readers. Therefore, the author should adhere to the guidelines for equations, units, and mathematical notation, available from various resources .

Protocols for mathematics writing get very specific – fonts, punctuation, examples, footnotes, sentences, paragraphs, and the title, all have detailed constraints and conventions applied to their usage. The American Mathematical Society is a good resource for additional guidelines.

## LaTeX and Wolfram

Mathematical sentences contain equations, figures, and notations that are difficult to typeset using a typical word-processing program. Both LaTeX and Wolfram have expert typesetting capabilities to assist authors in writing.

LaTeX is highly recommended for researchers whose papers constitute mathematical figures and notation. It produces professional-looking documents and authentically represents mathematical language.

Wolfram Language & System Documentation Center’s Mathematica has sophisticated and convenient mathematical typesetting technology that produces professional-looking documents.

The main differences between the two systems are due to cost and accessibility. LaTeX is freely available, whereas Wolfram is not. In addition, any updates in Mathematica will come with an additional charge. LaTeX is an open-source system, but Mathematica is closed-source.

## Good Writing and Logical Constructions

Regardless of the document preparation system selected, publication of a mathematics paper is similar to the publication of any academic research in that it requires good writing. Authors must apply a strict, logical construct when writing a mathematics research paper.

There are resources that provide very specific guidelines related to following sections to write and publish a mathematics research paper.

- Concept of a math paper
- Title, acknowledgment, and list of authors
- Introduction
- Body of the work
- Conclusion, appendix, and references
- Publication of a math paper
- Preprint archive
- Choice of the journal, submission
- Publication

The critical elements of a mathematics research paper are good writing and a logical construct that allows the reader to follow a clear path to the author’s conclusions.

Good advice. For me, writing an essay on mathematics was very difficult. I did not have enough time and knowledge to write a quality essay. I worked a lot in the library and read many articles on the Internet. I studied information about essay writing. But I couldn’t finish the essay in full. I had to look for professional writers on the subject of mathematics. He helped me finish a few paragraphs. The work was delivered on time and on an excellent assessment.

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## You are here

Writing math research papers: a guide for students and instructors.

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## Robert Gerver

- Table of Contents

Writing Math Research Papers is primarily a guide for high school students that describes how to write aand present mathematics research papers. But it’s really much more than that: it’s a systematic presentation of a philosophy that writing about math helps many students to understand it, and a practical method to move students from the relatively passive role of someone doing what is assigned to them, to creative thinkers and published writers who contribute to the mathematical literature.

As experienced writers know, the actual writing is not the half of it. William Zinsser once taught a writing class at the New School for Social Research which involved no writing at all: students talked through their ideas in class and through that process discovered the real story which could be written from their tangle of experiences, hopes and dreams. The actual writing was secondary, once they understood how to find the story and organize it.

Gerver, an experienced high school mathematics teacher, takes a similar approach. The primary audience is high school students who want to prepare formal papers or presentations, for contests or for a “math day” at their high school. But the discovery, research and organizational processes involved in writing an original paper, as opposed to rehashing information from a reference book, can help any student learn and understand math, and the experience will be useful even if the paper is never written.

Gerver leads students through a discovery process beginning with examining their own knowledge of mathematics and reviewing the basics of problem solving. The “math annotation” project follows next, in which students organize their class notes for one topic for presentation to their peers, resulting in a product similar to a section of a textbook or handbook, complete with illustrations and the necessary background and review material. Practical advice about finding a topic, developing it by keeping a research journal, and creating a final product, either a research paper or oral presentation, follows.

Writing Math Research Papers is directed primarily to students, and could be assigned as a supplementary textbook for high school mathematics classes. It will also be useful to teachers who incorporate writing into their classes or who serve as mentors to the math club, and for student teachers in similar situations. An appendix for teachers includes practical advice about helping students through the research and writing process, organizing consultations, and grading the student papers and presentations. Excerpts from student research papers are included as well, and more materials are available from the web site www.keypress.com/wmrp .

Robert Gerver, PhD, is a mathematics instructor at North Shore High School in New York. He received the Presidential Award for Excellence in Mathematical Teaching in 1988 and the Tandy Prize and Chevron Best Practices Award in Education in 1997. He has been publishing mathematics. Dr. Gerver has written eleven mathematics textbooks and numerous articles, and holds two U.S. patents for educational devices.

Sarah Boslaugh, ( [email protected] ) is a Performance Review Analyst for BJC HealthCare and an Adjunct Instructor in the Washington University School of Medicine, both in St. Louis, MO. Her books include An Intermediate Guide to SPSS Programming: Using Syntax for Data Management (Sage, 2004), Secondary Data Sources for Public Health: A Practical Guide (Cambridge, 2007), and Statistics in a Nutshell (O'Reilly, forthcoming), and she is Editor-in-Chief of The Encyclopedia of Epidemiology (Sage, forthcoming).

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8. Appendices

In the appendices you should include any data or material that supported your research but that was too long to include in the body of your paper. Materials in an appendix should be referenced at some point in the body of the report.

Some examples:

• If you wrote a computer program to generate more data than you could produce by hand, you should include the code and some sample output.

• If you collected statistical data using a survey, include a copy of the survey.

• If you have lengthy tables of numbers that you do not want to include in the body of your report, you can put them in an appendix.

## Sample Write-Up

Seating unfriendly customers, a combinatorics problem.

By Lisa Honeyman February 12, 2002

## The Problem

In a certain coffee shop, the customers are grouchy in the early morning and none of them wishes to sit next to another at the counter.

1. Suppose there are ten seats at the counter. How many different ways can three early morning customers sit at the counter so that no one sits next to anyone else?

2. What if there are n seats at the counter?

3. What if we change the number of customers?

4. What if, instead of a counter, there was a round table and people refused to sit next to each other?

## Assumptions

I am assuming that the order in which the people sit matters. So, if three people occupy the first, third and fifth seats, there are actually 6 (3!) different ways they can do this. I will explain more thoroughly in the body of my report.

## Body of the Report

At first there are 10 seats available for the 3 people to sit in. But once the first person sits down, that limits where the second person can sit. Not only can’t he sit in the now-occupied seat, he can’t sit next to it either. What confused me at first was that if the first person sat at one of the ends, then there were 8 seats left for the second person to chose from. But if the 1 st person sat somewhere else, there were only 7 remaining seats available for the second person. I decided to look for patterns. By starting with a smaller number of seats, I was able to count the possibilities more easily. I was hoping to find a pattern so I could predict how many ways the 10 people could sit without actually trying to count them all. I realized that the smallest number of seats I could have would be 5. Anything less wouldn’t work because people would have to sit next to each other. So, I started with 5 seats. I called the customers A, B, and C.

With 5 seats there is only one configuration that works.

As I said in my assumptions section, I thought that the order in which the people sit is important. Maybe one person prefers to sit near the coffee maker or by the door. These would be different, so I decided to take into account the different possible ways these 3 people could occupy the 3 seats shown above. I know that ABC can be arranged in 3! = 6 ways. (ABC, ACB, BAC, BCA, CAB, CBA). So there are 6 ways to arrange 3 people in 5 seats with spaces between them. But, there is only one configuration of seats that can be used. (The 1 st , 3 rd , and 5 th ).

Next, I tried 6 seats. I used a systematic approach to show that there are 4 possible arrangements of seats. This is how my systematic approach works:

Assign person A to the 1 st seat. Put person B in the 3 rd seat, because he can’t sit next to person A. Now, person C can sit in either the 5 th or 6 th positions. (see the top two rows in the chart, below.) Next suppose that person B sits in the 4 th seat (the next possible one to the right.) That leaves only the 6 th seat free for person C. (see row 3, below.) These are all the possible ways for the people to sit if the 1 st seat is used. Now put person A in the 2 nd seat and person B in the 4 th . There is only one place where person C can sit, and that’s in the 6 th position. (see row 4, below.) There are no other ways to seat the three people if person A sits in the 2 nd seat. So, now we try putting person A in the 3 rd seat. If we do that, there are only 4 seats that can be used, but we know that we need at least 5, so there are no more possibilities.

Possible seats 3 people could occupy if there are 6 seats

Once again, the order the people sit in could be ABC, BAC, etc. so there are 4 * 6 = 24 ways for the 3 customers to sit in 6 seats with spaces between them.

I continued doing this, counting how many different groups of seats could be occupied by the three people using the systematic method I explained. Then I multiplied that number by 6 to account for the possible permutations of people in those seats. I created the following table of what I found.

Next I tried to come up with a formula. I decided to look for a formula using combinations or permutations. Since we are looking at 3 people, I decided to start by seeing what numbers I would get if I used n C 3 and n P 3 .

3 C 3 = 1 4 C 3 = 4 5 C 3 = 10 6 C 3 = 20

3 P 3 = 6 4 P 3 = 24 5 P 3 = 60 6 P 3 = 120

Surprisingly enough, these numbers matched the numbers I found in my table. However, the n in n P r and n C r seemed to be two less than the total # of seats I was investigating.

## Conjecture 1:

Given n seats at a lunch counter, there are n -2 C 3 ways to select the three seats in which the customers will sit such that no customer sits next to another one. There are n -2 P 3 ways to seat the 3 customers in such a way than none sits next to another.

After I found a pattern, I tried to figure out why n -2 C 3 works. (If the formula worked when order didn’t matter it could be easily extended to when the order did, but the numbers are smaller and easier to work with when looking at combinations rather than permutations.)

In order to prove Conjecture 1 convincingly, I need to show two things:

(1) Each n – 2 seat choice leads to a legal n seat configuration.

(2) Each n seat choice resulted from a unique n – 2 seat configuration.

To prove these two things I will show

And then conclude that these two procedures are both functions and therefore 1—1.

Claim (1): Each ( n – 2) -seat choice leads to a legal n seat configuration.

Suppose there were only n – 2 seats to begin with. First we pick three of them in which to put people, without regard to whether or not they sit next to each other. But, in order to guarantee that they don’t end up next to another person, we introduce an empty chair to the right of each of the first two people. It would look like this:

We don’t need a third “new” seat because once the person who is farthest to the right sits down, there are no more customers to seat. So, we started with n – 2 chairs but added two for a total of n chairs. Anyone entering the restaurant after this procedure had been completed wouldn’t know that there had been fewer chairs before these people arrived and would just see three customers sitting at a counter with n chairs. This procedure guarantees that two people will not end up next to each other. Thus, each ( n – 2)-seat choice leads to a unique, legal n seat configuration.

Therefore, positions s 1 ' s 2 ', and s 3 ' are all separated by at least one vacant seat.

This is a function that maps each combination of 3 seats selected from n – 2 seats onto a unique arrangement of n seats with 3 separated customers. Therefore, it is invertible.

Claim (2): Each 10-seat choice has a unique 8-seat configuration.

Given a legal 10-seat configuration, each of the two left-most diners must have an open seat to his/her right. Remove it and you get a unique 8-seat arrangement. If, in the 10-seat setting, we have q 1 > q 2 , q 3 ; q 3 – 1 > q 2 , and q 2 – 1 > q 1 , then the 8 seat positions are q 1 ' = q 2 , q 2 ' = q 2 – 1, and q 3 ' = q 3 – 2. Combining these equations with the conditions we have

q 2 ' = q 2 – 1 which implies q 2 ' > q 1 = q 1 '

q 3 ' = q 3 – 2 which implies q 3 ' > q 2 – 1 = q 2 '

Since q 3 ' > q 2 ' > q 1 ', these seats are distinct. If the diners are seated in locations q 1 , q 2 , and q 3 (where q 3 – 1 > q 2 and q 2 – 1 > q 1 ) and we remove the two seats to the right of q 1 and q 2 , then we can see that the diners came from q 1 , q 2 – 1, and q 3 – 2. This is a function that maps a legal 10-seat configuration to a unique 8-seat configuration.

The size of a set can be abbreviated s ( ). I will use the abbreviation S to stand for n separated seats and N to stand for the n – 2 non-separated seats.

therefore s ( N ) = s ( S ).

Because the sets are the same size, these functions are 1—1.

Using the technique of taking away and adding empty chairs, I can extend the problem to include any number of customers. For example, if there were 4 customers and 10 seats there would be 7 C 4 = 35 different combinations of chairs to use and 7 P 4 = 840 ways for the customers to sit (including the fact that order matters). You can imagine that three of the ten seats would be introduced by three of the customers. So, there would only be 7 to start with.

In general, given n seats and c customers, we remove c- 1 chairs and select the seats for the c customers. This leads to the formula n -( c -1) C c = n - c +1 C c for the number of arrangements.

Once the number of combinations of seats is found, it is necessary to multiply by c ! to find the number of permutations. Looking at the situation of 3 customers and using a little algebraic manipulation, we get the n P 3 formula shown below.

This same algebraic manipulation works if you have c people rather than 3, resulting in n - c +1 P c

Answers to Questions

- With 10 seats there are 8 P 3 = 336 ways to seat the 3 people.
- My formula for n seats and 3 customers is: n -2 P 3 .
- My general formula for n seats and c customers, is: n -( c -1) P c = n - c +1 P c

_________________________________________________________________ _

After I finished looking at this question as it applied to people sitting in a row of chairs at a counter, I considered the last question, which asked would happen if there were a round table with people sitting, as before, always with at least one chair between them.

I went back to my original idea about each person dragging in an extra chair that she places to her right, barring anyone else from sitting there. There is no end seat, so even the last person needs to bring an extra chair because he might sit to the left of someone who has already been seated. So, if there were 3 people there would be 7 seats for them to choose from and 3 extra chairs that no one would be allowed to sit in. By this reasoning, there would be 7 C 3 = 35 possible configurations of chairs to choose and 7 P 3 = 840 ways for 3 unfriendly people to sit at a round table.

Conjecture 2: Given 3 customers and n seats there are n -3 C 3 possible groups of 3 chairs which can be used to seat these customers around a circular table in such a way that no one sits next to anyone else.

My first attempt at a proof: To test this conjecture I started by listing the first few numbers generated by my formula:

When n = 6 6-3 C 3 = 3 C 3 = 1

When n = 7 7-3 C 3 = 4 C 3 = 4

When n = 8 8-3 C 3 = 5 C 3 = 10

When n = 9 9-3 C 3 = 6 C 3 = 20

Then I started to systematically count the first few numbers of groups of possible seats. I got the numbers shown in the following table. The numbers do not agree, so something is wrong — probably my conjecture!

I looked at a circular table with 8 people and tried to figure out the reason this formula doesn’t work. If we remove 3 seats (leaving 5) there are 10 ways to select 3 of the 5 remaining chairs. ( 5 C 3 ).

The circular table at the left in the figure below shows the n – 3 (in this case 5) possible chairs from which 3 will be randomly chosen. The arrows point to where the person who selects that chair could end up. For example, if chair A is selected, that person will definitely end up in seat #1 at the table with 8 seats. If chair B is selected but chair A is not, then seat 2 will end up occupied. However, if chair A and B are selected, then the person who chose chair B will end up in seat 3 . The arrows show all the possible seats in which a person who chose a particular chair could end. Notice that it is impossible for seat #8 to be occupied. This is why the formula 5 C 3 doesn’t work. It does not allow all seats at the table of 8 to be chosen.

The difference is that in the row-of-chairs-at-a-counter problem there is a definite “starting point” and “ending point.” The first chair can be identified as the one farthest to the left, and the last one as the one farthest to the right. These seats are unique because the “starting point” has no seat to the left of it and the “ending point” has no seat to its right. In a circle, it is not so easy.

Using finite differences I was able to find a formula that generates the correct numbers:

Proof: We need to establish a “starting point.” This could be any of the n seats. So, we select one and seat person A in that seat. Person B cannot sit on this person’s left (as he faces the table), so we must eliminate that as a possibility. Also, remove any 2 other chairs, leaving ( n – 4) possible seats where the second person can sit. Select another seat and put person B in it. Now, select any other seat from the ( n – 5) remaining seats and put person C in that. Finally, take the two seats that were previously removed and put one to the left of B and one to the left of C.

The following diagram should help make this procedure clear.

In a manner similar to the method I used in the row-of-chairs-at-a-counter problem, this could be proven more rigorously.

An Idea for Further Research:

Consider a grid of chairs in a classroom and a group of 3 very smelly people. No one wants to sit adjacent to anyone else. (There would be 9 empty seats around each person.) Suppose there are 16 chairs in a room with 4 rows and 4 columns. How many different ways could 3 people sit? What if there was a room with n rows and n columns? What if it had n rows and m columns?

References:

Abrams, Joshua. Education Development Center, Newton, MA. December 2001 - February 2002. Conversations with my mathematics mentor.

Brown, Richard G. 1994. Advanced Mathematics . Evanston, Illinois. McDougal Littell Inc. pp. 578-591

## The Oral Presentation

Giving an oral presentation about your mathematics research can be very exciting! You have the opportunity to share what you have learned, answer questions about your project, and engage others in the topic you have been studying. After you finish doing your mathematics research, you may have the opportunity to present your work to a group of people such as your classmates, judges at a science fair or other type of contest, or educators at a conference. With some advance preparation, you can give a thoughtful, engaging talk that will leave your audience informed and excited about what you have done.

## Planning for Your Oral Presentation

In most situations, you will have a time limit of between 10 and 30 minutes in which to give your presentation. Based upon that limit, you must decide what to include in your talk. Come up with some good examples that will keep your audience engaged. Think about what vocabulary, explanations, and proofs are really necessary in order for people to understand your work. It is important to keep the information as simple as possible while accurately representing what you’ve done. It can be difficult for people to understand a lot of technical language or to follow a long proof during a talk. As you begin to plan, you may find it helpful to create an outline of the points you want to include. Then you can decide how best to make those points clear to your audience.

You must also consider who your audience is and where the presentation will take place. If you are going to give your presentation to a single judge while standing next to your project display, your presentation will be considerably different than if you are going to speak from the stage in an auditorium full of people! Consider the background of your audience as well. Is this a group of people that knows something about your topic area? Or, do you need to start with some very basic information in order for people to understand your work? If you can tailor your presentation to your audience, it will be much more satisfying for them and for you.

No matter where you are presenting your speech and for whom, the structure of your presentation is very important. There is an old bit of advice about public speaking that goes something like this: “Tell em what you’re gonna tell ’em. Tell ’em. Then tell ’em what you told ’em.” If you use this advice, your audience will find it very easy to follow your presentation. Get the attention of the audience and tell them what you are going to talk about, explain your research, and then following it up with a re-cap in the conclusion.

## Writing Your Introduction

Your introduction sets the stage for your entire presentation. The first 30 seconds of your speech will either capture the attention of your audience or let them know that a short nap is in order. You want to capture their attention. There are many different ways to start your speech. Some people like to tell a joke, some quote famous people, and others tell stories.

Here are a few examples of different types of openers.

You can use a quote from a famous person that is engaging and relevant to your topic. For example:

• Benjamin Disraeli once said, “There are three kinds of lies: lies, damn lies, and statistics.” Even though I am going to show you some statistics this morning, I promise I am not going to lie to you! Instead, . . .

• The famous mathematician, Paul Erdös, said, “A Mathematician is a machine for turning coffee into theorems.” Today I’m here to show you a great theorem that I discovered and proved during my mathematics research experience. And yes, I did drink a lot of coffee during the project!

• According to Stephen Hawking, “Equations are just the boring part of mathematics.” With all due respect to Dr. Hawking, I am here to convince you that he is wrong. Today I’m going to show you one equation that is not boring at all!

Some people like to tell a short story that leads into their discussion.

“Last summer I worked at a diner during the breakfast shift. There were 3 regular customers who came in between 6:00 and 6:15 every morning. If I tell you that you didn’t want to talk to these folks before they’ve had their first cup of coffee, you’ll get the idea of what they were like. In fact, these people never sat next to each other. That’s how grouchy they were! Well, their anti-social behavior led me to wonder, how many different ways could these three grouchy customers sit at the breakfast counter without sitting next to each other? Amazingly enough, my summer job serving coffee and eggs to grouchy folks in Boston led me to an interesting combinatorics problem that I am going to talk to you about today.”

A short joke related to your topic can be an engaging way to start your speech.

It has been said that there are three kinds of mathematicians: those who can count and those who can’t.

All joking aside, my mathematics research project involves counting. I have spent the past 8 weeks working on a combinatorics problem.. . .

To find quotes to use in introductions and conclusions try: http://www.quotationspage.com/

To find some mathematical quotes, consult the Mathematical Quotation Server: http://math.furman.edu/~mwoodard/mquot.html

To find some mathematical jokes, you can look at the “Profession Jokes” web site: http://www.geocities.com/CapeCanaveral/4661/projoke22.htm

There is a collection of math jokes compiled by the Canadian Mathematical Society at http://camel.math.ca/Recreation/

After you have the attention of your audience, you must introduce your research more formally. You might start with a statement of the problem that you investigated and what lead you to choose that topic. Then you might say something like this,

“Today I will demonstrate how I came to the conclusion that there are n ( n – 4)( n – 5) ways to seat 3 people at a circular table with n seats in such a way that no two people sit next to each other. In order to do this I will first explain how I came up with this formula and then I will show you how I proved it works. Finally, I will extend this result to tables with more than 3 people sitting at them.”

By providing a brief outline of your talk at the beginning and reminding people where you are in the speech while you are talking, you will be more effective in keeping the attention of your audience. It will also make it much easier for you to remember where you are in your speech as you are giving it.

## The Middle of Your Presentation

Because you only have a limited amount of time to present your work, you need to plan carefully. Decide what is most important about your project and what you want people to know when you are finished. Outline the steps that people need to follow in order to understand your research and then think carefully about how you will lead them through those steps. It may help to write your entire speech out in advance. Even if you choose not to memorize it and present it word for word, the act of writing will help you clarify your ideas. Some speakers like to display an outline of their talk throughout their entire presentation. That way, the audience always knows where they are in the presentation and the speaker can glance at it to remind him or herself what comes next.

An oral presentation must be structured differently than a written one because people can’t go back and “re-read” a complicated section when they are at a talk. You have to be extremely clear so that they can understand what you are saying the first time you say it. There is an acronym that some presenters like to remember as they prepare a talk: “KISS.” It means, “Keep It Simple, Student.” It may sound silly, but it is good advice. Keep your sentences short and try not to use too many complicated words. If you need to use technical language, be sure to define it carefully. If you feel that it is important to present a proof, remember that you need to keep things easy to understand. Rather than going through every step, discuss the main points and the conclusion. If you like, you can write out the entire proof and include it in a handout so that folks who are interested in the details can look at them later. Give lots of examples! Not only will examples make your talk more interesting, but they will also make it much easier for people to follow what you are saying.

It is useful to remember that when people have something to look at, it helps to hold their attention and makes it easier for them to understand what you are saying. Therefore, use lots of graphs and other visual materials to support your work. You can do this using posters, overhead transparencies, models, or anything else that helps make your explanations clear.

## Using Materials

As you plan for your presentation, consider what equipment or other materials you might want use. Find out what is available in advance so you don’t spend valuable time creating materials that you will not be able to use. Common equipment used in talks include an over-head projector, VCR, computer, or graphing calculator. Be sure you know how to operate any equipment that you plan to use. On the day of your talk, make sure everything is ready to go (software loaded, tape at the right starting point etc.) so that you don’t have “technical difficulties.”

Visual aides can be very useful in a presentation. (See Displaying Your Results for details about poster design.) If you are going to introduce new vocabulary, consider making a poster with the words and their meanings to display throughout your talk. If people forget what a term means while you are speaking, they can refer to the poster you have provided. (You could also write the words and meanings on a black/white board in advance.) If there are important equations that you would like to show, you can present them on an overhead transparency that you prepare prior to the talk. Minimize the amount you write on the board or on an overhead transparency during your presentation. It is not very engaging for the audience to sit watching while you write things down. Prepare all equations and materials in advance. If you don’t want to reveal all of what you have written on your transparency at once, you can cover up sections of your overhead with a piece of paper and slide it down the page as you move along in your talk. If you decide to use overhead transparencies, be sure to make the lettering large enough for your audience to read. It also helps to limit how much you put on your transparencies so they are not cluttered. Lastly, note that you can only project approximately half of a standard 8.5" by 11" page at any one time, so limit your information to displays of that size.

Presenters often create handouts to give to members of the audience. Handouts may include more information about the topic than the presenter has time to discuss, allowing listeners to learn more if they are interested. Handouts may also include exercises that you would like audience members to try, copies of complicated diagrams that you will display, and a list of resources where folks might find more information about your topic. Give your audience the handout before you begin to speak so you don’t have to stop in the middle of the talk to distribute it. In a handout you might include:

• A proof you would like to share, but you don’t have time to present entirely.

• Copies of important overhead transparencies that you use in your talk.

• Diagrams that you will display, but which may be too complicated for someone to copy down accurately.

• Resources that you think your audience members might find useful if they are interested in learning more about your topic.

## The Conclusion

Ending your speech is also very important. Your conclusion should leave the audience feeling satisfied that the presentation was complete. One effective way to conclude a speech is to review what you presented and then to tie back to your introduction. If you used the Disraeli quote in your introduction, you might end by saying something like,

I hope that my presentation today has convinced you that . . . Statistical analysis backs up the claims that I have made, but more importantly, . . . . And that’s no lie!

## Getting Ready

After you have written your speech and prepared your visuals, there is still work to be done.

- Prepare your notes on cards rather than full-size sheets of paper. Note cards will be less likely to block your face when you read from them. (They don’t flop around either.) Use a large font that is easy for you to read. Write notes to yourself on your notes. Remind yourself to smile or to look up. Mark when to show a particular slide, etc.
- Practice! Be sure you know your speech well enough that you can look up from your notes and make eye contact with your audience. Practice for other people and listen to their feedback.
- Time your speech in advance so that you are sure it is the right length. If necessary, cut or add some material and time yourself again until your speech meets the time requirements. Do not go over time!
- Anticipate questions and be sure you are prepared to answer them.
- Make a list of all materials that you will need so that you are sure you won’t forget anything.
- If you are planning to provide a handout, make a few extras.
- If you are going to write on a whiteboard or a blackboard, do it before starting your talk.

## The Delivery

How you deliver your speech is almost as important as what you say. If you are enthusiastic about your presentation, it is far more likely that your audience will be engaged. Never apologize for yourself. If you start out by saying that your presentation isn’t very good, why would anyone want to listen to it? Everything about how you present yourself will contribute to how well your presentation is received. Dress professionally. And don’t forget to smile!

Here are a few tips about delivery that you might find helpful.

- Make direct eye contact with members of your audience. Pick a person and speak an entire phrase before shifting your gaze to another person. Don’t just “scan” the audience. Try not to look over their heads or at the floor. Be sure to look at all parts of the room at some point during the speech so everyone feels included.
- Speak loudly enough for people to hear and slowly enough for them to follow what you are saying.
- Do not read your speech directly from your note cards or your paper. Be sure you know your speech well enough to make eye contact with your audience. Similarly, don’t read your talk directly off of transparencies.
- Avoid using distracting or repetitive hand gestures. Be careful not to wave your manuscript around as you speak.
- Move around the front of the room if possible. On the other hand, don’t pace around so much that it becomes distracting. (If you are speaking at a podium, you may not be able to move.)
- Keep technical language to a minimum. Explain any new vocabulary carefully and provide a visual aide for people to use as a reference if necessary.
- Be careful to avoid repetitive space-fillers and slang such as “umm”, “er”, “you know”, etc. If you need to pause to collect your thoughts, it is okay just to be silent for a moment. (You should ask your practice audiences to monitor this habit and let you know how you did).
- Leave time at the end of your speech so that the audience can ask questions.

Displaying Your Results

When you create a visual display of your work, it is important to capture and retain the attention of your audience. Entice people to come over and look at your work. Once they are there, make them want to stay to learn about what you have to tell them. There are a number of different formats you may use in creating your visual display, but the underlying principle is always the same: your work should be neat, well-organized, informative, and easy to read.

It is unlikely that you will be able to present your entire project on a single poster or display board. So, you will need to decide which are the most important parts to include. Don’t try to cram too much onto the poster. If you do, it may look crowded and be hard to read! The display should summarize your most important points and conclusions and allow the reader to come away with a good understanding of what you have done.

A good display board will have a catchy title that is easy to read from a distance. Each section of your display should be easily identifiable. You can create posters such as this by using headings and also by separating parts visually. Titles and headings can be carefully hand-lettered or created using a computer. It is very important to include lots of examples on your display. It speeds up people’s understanding and makes your presentation much more effective. The use of diagrams, charts, and graphs also makes your presentation much more interesting to view. Every diagram or chart should be clearly labeled. If you include photographs or drawings, be sure to write captions that explain what the reader is looking at.

In order to make your presentation look more appealing, you will probably want to use some color. However, you must be careful that the color does not become distracting. Avoid florescent colors, and avoid using so many different colors that your display looks like a patch-work quilt. You want your presentation to be eye-catching, but you also want it to look professional.

People should be able to read your work easily, so use a reasonably large font for your text. (14 point is a recommended minimum.) Avoid writing in all-capitals because that is much harder to read than regular text. It is also a good idea to limit the number of different fonts you use on your display. Too many different fonts can make your poster look disorganized.

Notice how each section on the sample poster is defined by the use of a heading and how the various parts of the presentation are displayed on white rectangles. (Some of the rectangles are blank, but they would also have text or graphics on them in a real presentation.) Section titles were made with pale green paper mounted on red paper to create a boarder. Color was used in the diagrams to make them more eye-catching. This poster would be suitable for hanging on a bulletin board.

If you are planning to use a poster, such as this, as a visual aid during an oral presentation, you might consider backing your poster with foam-core board or corrugated cardboard. A strong board will not flop around while you are trying to show it to your audience. You can also stand a stiff board on an easel or the tray of a classroom blackboard or whiteboard so that your hands will be free during your talk. If you use a poster as a display during an oral presentation, you will need to make the text visible for your audience. You can create a hand-out or you can make overhead transparencies of the important parts. If you use overhead transparencies, be sure to use lettering that is large enough to be read at a distance when the text is projected.

If you are preparing your display for a science fair, you will probably want to use a presentation board that can be set up on a table. You can buy a pre-made presentation board at an office supply or art store or you can create one yourself using foam-core board. With a presentation board, you can often use the space created by the sides of the board by placing a copy of your report or other objects that you would like people to be able to look at there. In the illustration, a black trapezoid was cut out of foam-core board and placed on the table to make the entire display look more unified. Although the text is not shown in the various rectangles in this example, you will present your information in spaces such as these.

Don’t forget to put your name on your poster or display board. And, don’t forget to carefully proof-read your work. There should be no spelling, grammatical or typing mistakes on your project. If your display is not put together well, it may make people wonder about the quality of the work you did on the rest of your project.

For more information about creating posters for science fair competitions, see

http://school.discovery.com/sciencefaircentral/scifairstudio/handbook/display.html ,

http://www.siemens-foundation.org/science/poster_guidelines.htm ,

Robert Gerver’s book, Writing Math Research Papers , (published by Key Curriculum Press) has an excellent section about doing oral presentations and making posters, complete with many examples.

## References Used

American Psychological Association . Electronic reference formats recommended by the American Psychological Association . (2000, August 22). Washington, DC: American Psychological Association. Retrieved October 6, 2000, from the World Wide Web: http://www.apastyle.org/elecsource.html

Bridgewater State College. (1998, August 5 ). APA Style: Sample Bibliographic Entries (4th ed) . Bridgewater, MA: Clement C. Maxwell Library. Retrieved December 20, 2001, from the World Wide Web: http://www.bridgew.edu/dept/maxwell/apa.htm

Crannell, Annalisa. (1994). A Guide to Writing in Mathematics Classes . Franklin & Marshall College. Retrieved January 2, 2002, from the World Wide Web: http://www.fandm.edu/Departments/Mathematics/writing_in_math/guide.html

Gerver, Robert. 1997. Writing Math Research Papers . Berkeley, CA: Key Curriculum Press.

Moncur, Michael. (1994-2002 ). The Quotations Page . Retrieved April 9, 2002, from the World Wide Web: http://www.quotationspage.com/

Public Speaking -- Be the Best You Can Be . (2002). Landover, Hills, MD: Advanced Public Speaking Institute. Retrieved April 9, 2002, from the World Wide Web: http://www.public-speaking.org/

Recreational Mathematics. (1988) Ottawa, Ontario, Canada: Canadian Mathematical Society. Retrieved April 9, 2002, from the World Wide Web: http://camel.math.ca/Recreation/

Shay, David. (1996). Profession Jokes — Mathematicians. Retrieved April 5, 2001, from the World Wide Web: http://www.geocities.com/CapeCanaveral/4661/projoke22.htm

Sieman’s Foundation. (2001). Judging Guidelines — Poster . Retrieved April 9, 2002, from the World Wide Web: http://www.siemens-foundation.org/science/poster_guidelines.htm ,

VanCleave, Janice. (1997). Science Fair Handbook. Discovery.com. Retrieved April 9, 2002, from the World Wide Web: http://school.discovery.com/sciencefaircentral/scifairstudio/handbook/display.html ,

Woodward, Mark. (2000) . The Mathematical Quotations Server . Furman University. Greenville, SC. Retrieved April 9, 2002, from the World Wide Web: http://math.furman.edu/~mwoodard/mquot.html

Making Mathematics Home | Mathematics Projects | Students | Teachers | Mentors | Parents | Hard Math Café |

## How to do Research on Mathematics

## Selected Subject Headings

Listed below is a sample of a few broad Library of Congress subject headings—made up of one word or more representing concepts under which all library holdings are divided and subdivided by subject—which you can search under and use as subject terms as well when searching online library catalogs for preliminary and/or additional research, such as books, audio and video recordings, and other references, related to your research paper topic. When researching materials on your topic, subject heading searching may be more productive than searching using simple keywords. However, keyword searching when using the right search method (Boolean, etc.) and combination of words can be equally effective in finding materials more closely relevant to the topic of your research paper.

## Academic Writing, Editing, Proofreading, And Problem Solving Services

Get 10% off with 24start discount code, suggested research topics in math.

- Business Mathematics
- Game Theory
- Mathematics—Philosophy
- Women in Mathematics

## Selected Keyword Search Strategies and Guides

Most online library indexes and abstracts and full-text article databases offer basic and advanced “keyword” searching of virtually every subject. In this case, combine keyword terms that best define your thesis question or topic using the Boolean search method (employing “and” or “or”) to find research most suitable to your research paper topic.

If your topic is “the importance of mathematics in the world,” for example, enter “importance” and “mathematics” with “and” on the same line to locate sources directly compatible with the primary focus of your paper. To find research on more specific aspects of your topic, alternate with one new keyword at a time with “and” in between (for example, “advancements and mathematics,” “contributions and mathematics,” “influence and mathematics,” etc.).

For additional help with keyword searching, navigation or user guides for online indexes and databases by many leading providers—including Cambridge Scientific Abstracts, EBSCO, H.W. Wilson, OCLC, Ovid Technologies, ProQuest, and Thomson Gale—are posted with direct links on library Web sites to guides providing specific instruction to using whichever database you want to search. They provide additional guidance on how to customize and maximize your search, including advanced searching techniques and grouping of words and phrases using the Boolean search method—of your topic, of bibliographic records, and of full-text articles, and other documents related to the subject of your research paper.

## Selected Source and Subject Guides

Guide to Information Sources in Mathematics and Statistics , by Martha A. Tucker and Nancy D. Anderson, 348 pages (Westport, Conn.: Libraries Unlimited, 2004)

Mathematics Education Research: A Guide for the Research Mathematician , by Curtis McKnight et al., 106 pages (Providence, R.I.: American Mathematical Society, 2000)

In addition to these sources of research, most college and university libraries offer online subject guides arranged by subject on the library’s Web page; others also list searchable course-related “LibGuides” by subject. Each guide lists more recommended published and Web sources—including books and references, journal, newspaper and magazines indexes, full-text article databases, Web sites, and even research tutorials—that you can access to expand your research on more specific issues and relevant to your subject.

## Selected Books and References

Dictionaries.

The Concise Oxford Dictionary of Mathematics , by Christopher Clapham and James Nicholson, 4th ed., 528 pages (Oxford and New York: Oxford University Press, 2009)

This revised fourth edition of applied mathematics and statistics features more than 3,000 entries, arranged in alphabetical order and illustrated with charts, diagrams, and graphs, covering technical mathematical terms, from Achilles paradox to zero matrix. Includes free access to regularly updated online version of the book.

Encyclopedic Dictionary of Mathematics , 2nd ed., edited by Mathematical Society of Japan and Kiyosi Ito, 4 vols. (Cambridge, Mass.: MIT Press, 1987)

This unique four-volume encyclopedia of applied mathematics features 450 articles, including 70 new articles since its first edition, published in 1977, covering such categories as algebra; group theory; number theory; Euclidean and projective geometry; differential geometry; algebraic geometry; topology; analysis; complex analysis; functional analysis; differential, integral, and functional equations; special functions; numerical analysis; computer science and combinatorics; probability theory; statistics; mathematical programming and operations research; mechanics and theoretical physics; and the history of mathematics.

The Facts on File Dictionary of Mathematics , 4th ed., by John Daintith and Richard Rennie, 262 pages (New York: Checkmark Books, 2005)

This dictionary covers mathematical terms and concepts—some 320 entries in all—fully illustrated, including lists of Web sites and bibliographies of sources.

The Penguin Dictionary of Mathematics , 4th ed., edited by David Nelson, 496 pages (London and New York: Penguin Books, 2008)

Everything from algebra to number theory and statistics to mechanics is thoroughly covered in this updated reference encompassing more than 3,200 cross-referenced entries from all branches of pure and applied mathematics. Also includes biographies of more than 200 major figures in mathematics.

## Encyclopedias

CRC Concise Encyclopedia of Mathematics , 2nd ed., by Eric W. Weisstein, 3,252 pages (Boca Raton, Fla.; London: Chapman & Hall/CRC, 2003)

This revised and expanded second edition—adding 1,000 pages of new illustrated material since its first edition—broadly covers mathematical definitions, formulas, figures, tabulations, and references on the subject.

Encyclopaedia of Mathematics , 11 vols., 5,400 pages (Dordrecht, Netherlands, and Boston: Reidel; Norwell, Mass.: Kluwer Academic Publishers, 1989–94; New York: Springer, 2005– )

This major unabridged 11-volume reference with index, hailed as “the most up-to-date, authoritative and comprehensive English-language work of reference in mathematics which exists today,” contains more than 7,000 cross-referenced entries covering all aspects of mathematics, including mathematical definitions, concepts, explanations, surveys, examples, terminology and methods, and more. In 2007, two new supplements—the first since the series was first published—were issued containing nearly 600 new entries in each written by experts in the field.

Encyclopedia of Mathematics Education , by Louise Grinstein and Sally I. Lipsey, 700 pages (New York: Routledge Falmer, 2001)

Designed for elementary, secondary, and post-secondary educators, this single-volume lists more than 400 alphabetically arranged entries covering all areas of mathematics education, including assessment, curriculum, enrichment, learning and instruction, and more.

Encyclopedia of Statistical Sciences , 2nd ed., edited by Samuel Kotz, et al., 16 vols., 9,686 pages (New York: Wiley, 2005)

Revised reference set expanded to 16 volumes and written by 600 experts detailing every area of statistical sciences, including its origin, new trends, and changes, and such areas as statistical theory and methods and application in biomedicine, computer science, economics, engineering, genetics, medicine, the environment, sociology, and more.

## Guides and Handbooks

Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences , 2nd ed., by Ivor Grattan-Guiness, 2 vols., 976 pages (Baltimore, Md.: Johns Hopkins University Press, 2003)

This illustrated two-volume set features 176 concise articles divided into 12 sections covering the development, history, cultural importance, problems, and theories and techniques of math and its execution in related sciences, including astronomy, computer science, engineering, philosophy, and social sciences, from its early beginnings through the 20th century. Features annotated bibliographies of sources with each article.

Figures of Thought: Mathematics and Mathematical Texts , by David Reed, 208 pages (London and New York: Routledge, 1994)

This single reference traces the history and evolution of mathematics and the work of famous mathematicians throughout history, including Dedekind, Descartes, Grothendieck, Hilbert, Kronecker, and Weil and an understanding of their approaches to mathematical science.

Guide to Information Sources in Mathematics and Statistics , by Martha A. Tucker and Nancy D. Anderson, 368 pages (Englewood, Colo.: Libraries Unlimited, 2004)

Praised as a “useful resource for college librarians and those just getting started in mathematics and statistics research,” this revised edition encompasses how to locate and access hundreds of print and electronic sources on mathematical sciences from 1800s to date.

A History of Mathematics: An Introduction , 3rd ed., by Victor J. Katz (Boston: Addison-Wesley, 2009)

This updated third edition provides a historical and world perspective of mathematics—its early and modern history, its evolving techniques, and contributions to the art of mathematics from throughout the Western and non-Western world.

INSTAT, International Statistics Sources: Subject Guide to Sources of International Comparative Statistics , by M. C. Fleming and J. G. Nellis, 1,080 pages (London: Routledge, 1994)

Unprecedented in its coverage, this A-to-Z guide details statistical data sources on business, economics, and social sciences topics, including agriculture, employment, energy, environment, finance, health, manufacturing, population, and wages. A subject index to all topics is included.

## Selected Full-Text Article Databases

ArticleFirst (Dublin, Ohio: OCLC FirstSearch, 1990– )

Full-text articles and citations to more than 16,000 journals in all subjects, including science, technology, and others; also known as OCLC ArticleFirst Database.

ESBCOHost Academic Search Elite (Ipswich, Mass.: ESBSCO Publishing, EBSCOHost, abstracting/indexing: 1984– , full text: 1990– )

A Web index of full-text articles from more than 1,250 journals, plus abstracts and citations from 3,200 journals covering general science, the social sciences, and more.

JSTOR (Ann Arbor, Mich.: Journal Storage Project, 1800s—latest 3 to 5 years)

A Web archive of important scholarly journals, some in full text, including more than 37,000 articles from The American Mathematical Monthly (1894–2004), since the 1800s in economics, finance, and mathematical sciences.

ScienceDirect (St. Louis, Mo.: Elsevier Science, 1995– )

Leading science database on the Web with full-text access to more than 9.5 million articles from more than 2,500 scientific, mathematical, technical, and social science journals.

Web of Science (Philadelphia: Thomson Scientific, 1840– )

Contains detailed bibliographic records to more than 8,700 worldwide scientific journals and publications, with full-text articles from more than 250 scientific journals from 1840 to date; allows searching of material from other related databases, including Science Citation Index (1900– ), Social Sciences Citation Index (1956– ), Arts & Humanities Citation Index (1975– ), Index Chemicus (1993– ), and Current Chemical Reactions (1986– ).

Wilson Select Plus (Bronx, N.Y.: H.W. Wilson Co., WilsonDisc/OCLC FirstSearch, 1994– )

On the Web, indexes and abstracts full-text articles from 2,621 journals, magazines, and newspapers covering such subjects as science, humanities, education, and business.

## Selected Periodicals

Acta Mathematica Sinica (Tokyo, Japan: Springer-Verlag Tokyo/Chinese Mathematical Society, 1936– )

This English-translated version of the popular quarterly journal published by the Chinese Mathematical Society since 1936 (originally titled, Journal of Chinese Mathematical Society, until it was renamed in 1952) publishes authoritative reviews of current citations with abstracts to articles from current and past issues searchable in such online databases as Academic OneFile, Current Abstracts, Current Contents/Physical, Chemical and Earth Sciences, International Abstracts in Operations Research, Journal Citation Reports/Science Edition, Mathematical Reviews, Science and Technology Collection, Science Citation Index Expanded, SCOPUS, TOC Premier, and Zentralblatt MATH.

American Journal of Mathematics (Baltimore, Md.: Johns Hopkins University Press, 1878– )

As “the oldest mathematics journal in the Western Hemisphere,” published since 1878, this academic journal, one of the most respected and celebrated in its industry, publishes pioneering mathematical papers and articles about all areas of contemporary mathematics. Articles are indexed and abstracted in the following electronic databases: CompuMath Citation Index, Current Contents/Physical, Chemical and Earth Sciences, Current Mathematical Publications, General Science Index, Index to Scientific Reviews, Math-SciNet, Mathematical Reviews, Science Citation Index, Social Science Citation Index, and Zentralblatt MATH. To browse journals by subject or title, or search past issues, visit http://www.press.jhu.edu/journals/american_journal_of_mathematics/ .

Annals of Applied Probability (Beachwood, Ohio: Institute of Mathematical Statistics, February 1991– )

First published in February 1991, this scholarly journal publishes important and original research covering all facets of contemporary applications of probability. Electronic access to all issues of the journal is available through JSTOR (issues older than 3 years from the current year).

Annals of Combinatorics (Singapore and New York: Springer-Verlag, 1997– )

This journal covers new developments, mathematical breakthroughs and mathematical theories in combinatorial mathematics, particularly its applications to computer science, biology, statistics, probability, physics, and chemistry, as well as representation theory, number theory topology, algebraic geometry, and more. Articles are indexed and fully searchable in Academic OneFile, Current Abstracts, Current Contents/Physical, Chemical and Earth Sciences, Journal Citation Reports/Science Edition, Mathematical Reviews, Science and Technology Collection, Science Citation Index Expanded, and others.

Foundations of Computational Mathematics (New York: Springer-Verlag New York, 2001– )

Introduced in January 2001, this quarterly academic journal, published in association with the Foundations of Computational Mathematics, features articles discussing the connections between mathematics and computation, including the interfaces between pure and applied mathematics, numerical analysis, and computer science. Full-text articles from issues since 2001 can be viewed in PDF form at http://link.springer.com/journal/10208 .

Historia Mathematica (Amsterdam, Netherlands: Elsevier Science B.V., 1974– )

Launched in February 1974, this quarterly periodical of the International Commission on the History of Mathematics of the Division of the History of Science of the International Union of the History and Philosophy of Science covers all aspects of mathematical sciences, including mathematicians and their work, organizations and institutions, pure and applied mathematics, and the sociology of mathematics, as well as all cultures and historical periods of mathematics and its development. The primary aim and focus of each issue is topics in the history of math, including research articles, book reviews, and more. Full-text articles are accessible through ScienceDirect.

Journal of Mathematics and Statistics (New York: Science Publications, 2005– )

Published since January/March 2005, this peer-reviewed, open-access international scientific journal presents original and valuable research in all areas of applied and theoretical mathematics and statistics. To view or search back issues, visit http://thescipub.com/jmss.toc .

Journal of Pure and Applied Algebra (Evanston, Ill.: Elsevier Science, 1971– )

This monthly journal published in association with Northwestern University’s math department focuses on the development and theories of pure and applied algebra. Also available in microform since its first issue in January 1971, full-text articles from all issues can be searched in Elsevier Science’s ScienceDirect online database.

The Journal of Symbolic Logic (Poughkeepsie, N.Y.: Association for Symbolic Logic, 1936– )

Leading scientific journal founded in 1936 by the Association for Symbolic Logic (ASL) containing scholarly work and research on symbolic logic. The journal is distributed with two others published by the ASL: The Bulletin of Symbolic Logic and Review of Symbolic Logic. Full-text access to The Journal of Symbolic Logic (1936–2003) is available via JSTOR.

Journal of the American Mathematical Society (JAMS) (Providence, R.I.: American Mathematical Society, January 1, 1988– )

Mathematics journal published quarterly by the American Mathematical Society reporting research in all areas of pure and applied mathematics. Journal articles are indexed in such subscription Web databases as Citation Index—Expanded, CompuMath Citation Index, and Current Contents, Physical, Chemical & Earth Sciences. Since January 1996, JAMS is also accessible online at http://www.ams.org/publications/journals/journalsframework/jams .

Mathematical Physics, Analysis, and Geometry (Norwell, Mass.: Kluwer Academic/Plenum Publishing Corp., 1998– )

Scientific journal covering concrete problems of mathematics and theoretical analysis and application of analysis on all math, from geometry to physics, including problems of statistical physics and fl uids; complex function theory; operators in function space, especially operator algebras; ordinary and partial differential equations; and differential and algebraic geometry. Journal is indexed and abstracted in many subject-specific online databases, including Academic OneFile, Current Abstracts, Google Scholar, Journal Citation Reports/Science Edition, Mathematical Reviews, and others.

SIAM Journal on Applied Mathematics (Newark, Dela.: Society for Industrial & Applied Mathematics, 1953– )

Published by the Society for Industrial & Applied Mathematics since 1953, this quarterly journal reviews applied mathematics of physical, engineering, biological, medical, and social sciences, including research articles discussing problems and methods pertinent to physical, engineering, financial, and life sciences. Full bibliographic records with abstracts of articles from 1997 to the present can be searched on SIAMS Journals Online at http://www.siam.org/journals/siap.php .

## Selected Web Sites

American Mathematical Society ( http://www.ams.org/home/page )

Association of professional mathematicians, headquartered in Providence, Rhode Island, reporting on mathematical research and education, conferences, surveys, publications, scholarship programs, and more.

American Statistical Association ( http://www.amstat.org/ )

This Web site for the nation’s leading professional organization for statisticians and professors provides resources for visitors and members, including association news, membership information, educational opportunities, publications, meetings and events, and outreach programs.

Euler Archive—Dartmouth College ( http://eulerarchive.maa.org/ )

Provides online access to Dartmouth College’s archive of 866 original works of pioneering Swiss mathematician Leonard Euler as well as other original publications and current research.

MacTutor History of Mathematics ( http://www-history.mcs.st-andrews.ac.uk/history/index.html )

Features searchable historical mathematical topics, biographies of notable mathematicians from AD 500 to present, and an index of famous curves.

Math Archives ( http://archives.math.utk.edu/ )

Online resource covering a wide range of mathematical topics and Internet resources arranged by subject.

Mathematical Atlas ( http://www.math-atlas.org/ )

Gateway collection of articles discussing various mathematical concepts, with links to additional resources in all areas of mathematics.

Mathematics on the Web ( http://www.mathontheweb.org/mathweb/ )

Online mathematical sources maintained by the American Mathematical Society and organized by subject, including article abstracts and databases, as well as bibliographies; books, journals, columns, and handbooks; math history and math topics; information about mathematics departments, institutes, centers, associations, societies, and organizations; and related software and tools.

Mathematics, Statistics, and Computational Science at NIST ( http://math.nist.gov/ )

Site provided by the National Institute of Standards and Technology, offering information on NIST projects, events, and organizations; math software; statistical guides and handbooks; statistical data sets; and more.

Math Forum ( http://mathforum.org/ )

This site, maintained by Drexel University’s School of Education, offers both resources and information on math and math education, along with access to the Internet Mathematics Library, discussion groups, and more.

Math on the Web ( http://www.mathontheweb.org/mathweb/index.html )

Web portal produced by the American Mathematical Society providing access to mathematical news and information, journals, and reference materials.

Math World ( http://mathworld.wolfram.com/ )

Deemed by its creators as “the Web’s most extensive mathematics resource,” this site includes a searchable math dictionary and encyclopedia, interactive tools, and information on the computational software program Mathematica.

+plus magazine ( http://plus.maths.org/content/ )

Free access to this weekly online magazine featuring the latest mathematical news, articles by leading mathematicians and science writers, a browsable archive, and information on a variety of mathematical applications.

Probability Tutorials ( http://www.probability.net/ )

Online math tutorials explaining probability, definitions, theorems, solutions, and more.

SIAM: Society for Industrial and Applied Mathematics ( http://www.siam.org/ )

Official Web site for the Society for Industrial and Applied Mathematics offering information on books, careers and jobs, conferences, journals, proceedings, and the latest news.

Zentralblatt MATH ( http://www.zentralblatt-math.org/zmath/en/ )

Searchable database of more than 2 million citations with abstracts to books, journals, conference reports, and more, from 1868 to present.

## Careers Related to Mathematics

Science, Technology, Engineering, and Mathematics Career Cluster ( http://career.iresearchnet.com/career-clusters/science-technology-engineering-and-mathematics-career-cluster/ )

Science careers include jobs in biology, chemistry, geology, meteorology, or any other natural, physical, or earth science. Mathematics is the science and study of numbers and how they relate to each other. Engineering and technology encompasses many areas of study, such as aviation, environmental science, and robotics, just to name a few. All of these engineering fields employ unique and sometimes similar methods of research, development, and production to reach practical solutions to problems and questions.

Mathematics and Physics Career Field ( http://career.iresearchnet.com/career-fields/mathematics-and-physics-career-field/ )

Mathematics and physics are closely related natural sciences. Mathematics is the science and study of numbers and how they relate with each other. Physics is the study of the basic elements and laws of the universe.

Engineering Career Field ( http://career.iresearchnet.com/career-fields/engineering-career-field/ )

A lot of brainpower goes into engineering—a lot of knowledge, creativity, thoughtfulness, and pure hard work. Humankind has been “engineering,” so to speak, since we realized we had opposable thumbs that we could use to handle tools. And from that point on we began our ceaseless quest to make, to build, to create tools and systems that helped us live our lives better. There were a lot of mistakes, but engineers and scientists learned from them and built a foundation of engineering laws and principles.

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Theorem 1.2.1. A homogenous system

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