what is a presentation in math

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After you have finished doing your mathematics research, you will need to present your findings to others. There are three main ways to do this:

The following sections provide information about each of these presentation strategies.

Your project write-up is a chance to synthesize what you have learned about your mathematics research problem and to share it with others. Most people find that when they complete their write-up it gives them quite a bit of satisfaction. The process of writing up research forces you to clarify your own thinking and to make sure you really have rigorous arguments. You may be surprised to discover how much more you will learn by summarizing your research experience!

If you have ever looked at a mathematics journal to see how mathematicians write up new results, you may have found that everything seemed neat and polished. The author often poses a question and then presents a proof that leads neatly, and sometimes elegantly, to his or her solution. Mathematicians rarely talk about the dead ends they met along the way in these formal presentations. Your write-up will be different. We hope that you will tell your reader about your thought process. How did you start? What did you discover? Where did that lead you? What were your conjectures? Did you disprove any of them? How did you prove the ones that were true? By answering these questions, you will provide a detailed map that will take the reader through your research experience.

This guide will give you a brief overview of the parts of a mathematics research paper. Following the guide is a sample write up so you can see how one person wrote about her research experience and shared her results.

A formal mathematics research paper includes a number of sections. These will be appropriate for your write-up as well. The sections of the report are linked so that you can see an example of each part in the sample write-up that follows. Note that not all mathematical research reports contain all of the sections. For example, you might not have any appendices to include or you may not do a literature review. However, your write-up should definitely contain parts 1, 2, 4, 5, 6 and 7.

1.

2.

3.

In the literature review section you may answer questions such as “What kind of research has been done before?” “What kind of relevant studies or techniques needed to be mastered to do your project?” “How have others gone about trying to solve your problem, and how does your approach differ?”

4.

5.

The body of your report should be a mix of English narrative and more abstract representations. Be sure to include lots of examples to help your reader understand your reasoning. If your paper is very long, you can divide the body of your report into sections so that it easier to tackle the various aspects of your work.

6.

7.

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 |

Browse Course Material

Course info, instructors.

• Prof. Haynes Miller
• Dr. Nat Stapleton
• Saul Glasman

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• Mathematics

As Taught In

Learning resource types, project laboratory in mathematics, presentations.

Next: Practice and Feedback »

In this section, Prof. Haynes Miller and Susan Ruff describe the criteria for presentations and the components of the presentation workshop.

Criteria for Good Presentations

Effective presentations provide motivation, communicate intuition, and stimulate interest, all while being mathematically accurate and informative. As is true with their experience with mathematical writing, many students do not enter the course in possession of the tools to do much more than present the facts. For example, students often come to practice presentations with the mistaken belief that a mathematical presentation must be extremely formal throughout, every term must be rigorously defined, all facts must be proven, and pictures are too infantile for this level of presentation. We try to counter these preconceptions and urge flexibility and a sense of appropriateness: sometimes things need to be presented rigorously and formally, but sometimes a picture, conceptual explanation, or example is much more effective.

Characteristics of an Effective Undergraduate Research Talk (PDF)

Presentation Workshop

For the presentation workshop, which typically lasts 50 to 80 minutes, we begin by having the two co-instructors each give a short mock presentation. These presentations are designed to address common student misconceptions about mathematics presentations. For example, to help students realize that presentations should not be relentlessly formal, the first presentation might be good in every way except that it is dull and difficult to follow because it is unnecessarily formal throughout. In contrast, the second presentation might cover the same material but use examples and figures to introduce some concepts informally, while reserving rigorous formality as a strategy for clarifying and solidifying the most subtle or important concepts.

To help students recognize the value of the second presentation relative to the first, after each presentation we ask the students a question designed to check their understanding of the content. The goal is to allow students to discover their natural tendency to overlook weaknesses in presentations. When they try to answer questions about it, they may discover that they got less from it than they had thought. The second presentation is then intended to offer a more understandable approach to the same material. Of course it’s the second time students will have heard this material, so they will naturally understand it better. But this serves a pedagogical purpose too, as it reinforces our point.

We follow the presentations with a class discussion on how to give a good presentation. Carefully designing two mock presentations has the virtue of drawing attention to key learning objectives, but doing so is challenging. In Spring 2013, each mock presentation was delivered by a different instructor and so had different advantages and disadvantages, as is stressed by Haynes’ comments on the workshop (PDF) . In the past we have reduced accidental differences between the presentations by having a single instructor present both, and we may return to that approach in the future.

After the mock presentations, the class discusses the characteristics of a good presentation. Questions we discuss often include the following:

• What are the reasons to include a proof in a presentation?
• What other strategies are available for achieving these goals?
• What strategies can be used to make a math presentation engaging for the target audience of math majors?

In Spring 2013, the mock presentations ran long, and the class session was shorter than we had originally planned because of scheduling disruptions at MIT. Thus, the subsequent discussion was rushed. The presentation workshop works best when there is ample time for discussion.

We hope that students come away from this workshop with an appreciation for some of the complexities in designing a good presentation. Pretty much every choice involved has both pros and cons.

• Download video

This video features the presentation workshop from Spring 2013. The co-instructors deliver mock presentations, which are followed by a brief class discussion comparing the two presentations.

Chalk Talks versus Slide Presentations

Different instructors have set different expectations for the presentations. Some have insisted on slide presentations. More typically, students are encouraged to use media suited to the demands of the presentation.

When discussing slide presentations in mathematics, we usually make the following points:

• When slides contain large amounts of text (or equations), the audience cannot read and listen at the same time, so strategies are needed either to reduce the content on the slides or to guide the audience through the content.
• The audience needs time to absorb math concepts, but it is very easy to click through slides too quickly, especially when the presenter is nervous, so strategies are needed to give the audience time to think.
• The audience cannot refer to past slides to remind themselves of the meaning of new notation or of the purpose of details being presented, so strategies are needed to help the audience remember important points.

A Note about Scheduling

In the course, roughly one group presents each week. Experience has shown that the first team to present sets the bar for the rest of the semester. It is important that the first team be chosen carefully and be guided well so that they give a strong presentation.

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How to Make A Breathtaking Mathematics Presentation

by admin | Aug 16, 2017 | Educational Process | 0 comments

Any mathematician knows how important presentations are to sharing the work. Meanwhile, a breathtaking mathematic presentation might sound as an oxymoron. Even if it seems as if the math presentation topics you choose are impressively compelling, the way you “sell” your information is vital.

Confused and yawning faces are not what you want to see when giving the presentation. These tips may help you get the best results out of your next performance.

Study Your Target Audience

In most cases, you know who your audience is and what their professional level, knowledge, and needs are. The complexity of your presentation must depend on who listens to it. Tailoring it to their needs ensures better understanding and effect. Is it high school or college students? Colleagues or researchers? Employees or bosses? Make it your goal to find out.

Reveal Your Goal Immediately

Before starting your presentation, let your audience know why it’s important for them. How can it help with their research? Which problems can it assist them in solving? How can this knowledge change their approach to a certain project? Perhaps it can help them get a new job or pass a math exam. Listening to a bunch of info before you know the goal takes half of effectiveness out of the presentation.

Show Your Passion

When a speaker is passionate about a subject, the audience is bound to catch the mood. You can come up with an anecdote or a life story that made you truly excited about the presentation topic. Personalization is the key to making the audience more interesting in what you have to say. No matter how exciting the bare facts seem to you, they usually fail to catch the audience’s attention.

Use Math Presentation Software

If you use PowerPoint math presentations, you are bound to convey more information to your audience in the shortest amount of time. Meanwhile, other presentation software, such as Prezi, can help you secure the audience’s attention faster and keep it piqued. Custom Prezi presentation design is exactly what you need to share most of the math-related topics in the most beneficial manner.

Keep It Simple

Even though math is filled with jargon and acronyms that you may love, others may not be as savvy as you are. Keeping the presentation as simple as possible is the smartest way to get your point through. When listeners stumble upon an unknown term, they may stop listening to the rest of your thought trying to figure out where they heard it before. If you want to get a hold of a person’s attention, keep the presentation as simple as possible.

Be Ready to Perform

Even if you don’t have stage fright, the presentation may be a tough task . Every math speaker has to turn into a performer to reach out to the audience and get the best out of the process. If you know your subject well and have sufficient tools ready, there isn’t much to be afraid of. Avoid memorizing your speech. It may make you sound too automated. Take advantage of the presentation software instead to enhance your performance.

Don’t Go Beyond The Timeframe

Keeping to the timeframe is compulsory to helping your presentation make the right impression. Once your time is up, people start being uneasy about staying in the room. This can make the rest of your presentation useless. Always leave enough time to answer the questions. If no questions arise, you can use the time to go back to one of the most important slides and discuss it further.

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How to present math in talks

Since writing my last post ( The cognitive style of better powerpoint ), I noticed that two other bloggers wrote rather recently on the same topic. The first, from Dave Munger at Cognitive Daily , actually proposes a bit of an experiment to compare the efficacy of text vs. powerpoint - results to be posted Friday. The second, from Chad Orzel at Uncertain Principles , offers a list of "rules of thumb" for doing a good PowerPoint talk.

Given all this, you'd think I wouldn't have anything to add, right? Well, never underestimate my willingness to blather on and on about something. I actually think there's one thing neither they nor I discuss much, and that is presenting mathematical, technical, or statistical information. Both Orzel and I recommend, as much as possible, avoiding equations and math in your slides. And that's all well and good, but sometimes you just have to include some (especially if you're a math teacher and the talk in question is a lecture). For me, this issue crops up whenever I need to describe a computational model -- you need to give enough detail that it doesn't look like the results just come out of thin air, because if you don't, nobody will care about what you've done. And often "enough detail" means equations.

So, for whatever it's worth, here are my suggestions for how to present math in the most painless and effective way possible:

Abandon slideware. This isn't always feasible (for instance, if the conference doesn't have blackboards), nor even necessarily a good idea if the equation count is low enough and the "pretty picture" count is high enough, but I think slideware is sometimes overused, especially if you're a teacher. When you do the work on the blackboard, the students do it with you; when you do it on slideware, they watch. It is almost impossible to be engaged (or keep up) when rows of equations appear on slides; when the teacher works out the math on the spot, it is hard not to. (Okay, hard er ).

If you can't abandon slideware:

1. Include an intuitive explanation of what the equation means. (This is a good test to make sure you understand it yourself!). Obviously you should always do this verbally, but I find it very useful to write that part in text on the slide also. It's helpful for people to refer to as they try to match it with the equation and puzzle out how it works and what it means -- or, for the people who aren't very math-literate, to still get the gist of the talk without understanding the equation at all.

2. Decompose the equation into its parts. This is really, really useful. One effective way to do this is to present the entire thing at once, and then go through each term piece-by-piece, visually "minimizing" the others as you do so (either grey them out or make them smaller). As a trivial example, consider the equation z = x/y. You might first grey out (y) and talk about x. Then talk about y and grey out x: you might note things like that y is the denominator, you can see that when y gets larger our result gets smaller, etc. My example is totally lame, but this sort of thing can be tremendously useful when you get equations that are more complicated. People obviously know what numerators and denominators are, but it's still valuable to explicitly point out in a talk how the behavior of your equation depends on its component parts -- people could probably figure it out given enough time, but they don't have that time, particularly when it's all presented in the context of loads of other new information. And if the equation is important enough to put up, it's important to make sure people understand all of its parts.

3. As Orzel mentioned, define your terms. When you go through the parts of the equation you should verbally do this anyway, but a little "cheat sheet" there on the slide is invaluable. I find it quite helpful sometimes to have a line next to the equation that translates the equation into pseudo-English by replacing the math with the terms. Using my silly example, that would be something like "understanding (z) = clarity of images (x) / number of equations (y)". This can't always be done without cluttering things too much, but when you can, it's great.

4. Show some graphs exploring the behavior of your equation. ("Notice that when you hold x steady, increasing y results in smaller z"). This may not be necessary if the equation is simple enough, but if it's simple enough maybe you shouldn't present it, and just mention it verbally or in English. If what you're presenting is an algorithm, try to display pictorially what it looks like to implement the algorithm. Also, step through it on a very simple dataset. People remember and understand pictures far better than equations most of the time.

5. When referring back to your equation later, speak English. By this I mean that if you have a variable y whose rough English meaning is "number of equations", whenever you talk about it later, refer to it as "number of equations", not y. Half of the people won't remember what y is after you move on, and you'll lose them. If you feel you must use the variable name, at least try to periodically give reminders about what it stands for.

6. Use LaTeX where possible . LaTeX's software creates equations that are clean and easy to read, unlike PowerPoint (even with lots of tweaking). You don't necessarily have to do the entire talk in LaTeX if you don't want to, but at least make the equations in LaTeX, screen capture them and save them as bitmaps, and paste them into PowerPoint. It is much, much easier to read.

Obviously, these points become more or less important depending on the mathematical sophistication of your audience, but I think it's far far easier to make mathematical talks too difficult rather than too simple. This is because it's not a matter (or not mainly a matter) of sophistication -- some of the most egregious violaters of these suggestions that I've seen have been at NIPS , a machine learning conference -- it's a matter of how much information your audience can process in a short amount of time. No matter how mathematically capable your listeners are, it takes a while (and a fair amount of concentration) to see the ramifications and implications of an equation or algorithm while simultaneously fitting it in with the rest of your talk, keeping track of your overall point, and thinking of how all of this fits in with their research. The easier you can make that process, the more successful the talk will be.

Any agreements, disagreements, or further suggestions, I'm all ears.

Posted by Amy Perfors at November 16, 2006 11:24 AM

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How do I prepare a slide talk presentation for conference in mathematics?

I am a graduate student in mathematics. I am preparing to give a 20-minute short talk for a conference. I am looking at some beamer presentations this one and this one . Neither of these slides neither have any references, nor any numbering of definitions, theorems, propositions, lemmas etc.

I have some few questions in mind:

• Is it normal to omit proof of the main results?
• Is it normal to omit numberings of theorems, lemmas, propositions, etc.?
• Why don't people use \begin{theorem}... \end{theorem} for theorems and the same for lemmas, and propositions?

Besides that, what would be the best way to present a short 20-minute conference talk?

• mathematics
• presentation

• 22 Your talk is only 20 minutes long. Putting proofs into a beamer talk most likely results in losing the audience, especially, if proofs use more than one slide and if the audience has to remember formulae from previous slides. Similar for numberings of theorems. –  Marktmeister Commented May 23, 2022 at 14:31
• 3 And just to complete the picture, I'll add to all the good advice that you've already gotten: Keep in mind that at some conferences there's even the option to use blackboards instead of slides. –  Jochen Glueck Commented May 23, 2022 at 20:37
• 2 I am thankful to all for giving me valuable suggestions. All the answers are just equally excellent. –  learner Commented May 24, 2022 at 2:53
• 3 The Notices of the American Mathematical Society has a column dedicated to advice for early-career scholars—I recommend checking it out for articles like this one by Bryna Kra that directly addresses your question. Indeed, you might ask your advisor to sponsor you for a student membership to the AMS and other professional societies. –  Greg Martin Commented May 24, 2022 at 7:24
• 4 As someone who works in a math department: in this field, you can get away with anything. Want yo just present your paper in beamer format, no changes to content? People do it. Want to have an extremely didactic (skipping proofs) presentation? people do it. There is no norm. My advice (from a non mathematician like me) is to try to get to the most people in the audience. –  Ander Biguri Commented May 24, 2022 at 10:24

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Presentation of Data

Statistics deals with the collection, presentation and analysis of the data, as well as drawing meaningful conclusions from the given data. Generally, the data can be classified into two different types, namely primary data and secondary data. If the information is collected by the investigator with a definite objective in their mind, then the data obtained is called the primary data. If the information is gathered from a source, which already had the information stored, then the data obtained is called secondary data. Once the data is collected, the presentation of data plays a major role in concluding the result. Here, we will discuss how to present the data with many solved examples.

What is Meant by Presentation of Data?

As soon as the data collection is over, the investigator needs to find a way of presenting the data in a meaningful, efficient and easily understood way to identify the main features of the data at a glance using a suitable presentation method. Generally, the data in the statistics can be presented in three different forms, such as textual method, tabular method and graphical method.

Presentation of Data Examples

Now, let us discuss how to present the data in a meaningful way with the help of examples.

Consider the marks given below, which are obtained by 10 students in Mathematics:

36, 55, 73, 95, 42, 60, 78, 25, 62, 75.

Find the range for the given data.

Given Data: 36, 55, 73, 95, 42, 60, 78, 25, 62, 75.

The data given is called the raw data.

First, arrange the data in the ascending order : 25, 36, 42, 55, 60, 62, 73, 75, 78, 95.

Therefore, the lowest mark is 25 and the highest mark is 95.

We know that the range of the data is the difference between the highest and the lowest value in the dataset.

Therefore, Range = 95-25 = 70.

Note: Presentation of data in ascending or descending order can be time-consuming if we have a larger number of observations in an experiment.

Now, let us discuss how to present the data if we have a comparatively more number of observations in an experiment.

Consider the marks obtained by 30 students in Mathematics subject (out of 100 marks)

10, 20, 36, 92, 95, 40, 50, 56, 60, 70, 92, 88, 80, 70, 72, 70, 36, 40, 36, 40, 92, 40, 50, 50, 56, 60, 70, 60, 60, 88.

In this example, the number of observations is larger compared to example 1. So, the presentation of data in ascending or descending order is a bit time-consuming. Hence, we can go for the method called ungrouped frequency distribution table or simply frequency distribution table . In this method, we can arrange the data in tabular form in terms of frequency.

For example, 3 students scored 50 marks. Hence, the frequency of 50 marks is 3. Now, let us construct the frequency distribution table for the given data.

Therefore, the presentation of data is given as below:

The following example shows the presentation of data for the larger number of observations in an experiment.

Consider the marks obtained by 100 students in a Mathematics subject (out of 100 marks)

95, 67, 28, 32, 65, 65, 69, 33, 98, 96,76, 42, 32, 38, 42, 40, 40, 69, 95, 92, 75, 83, 76, 83, 85, 62, 37, 65, 63, 42, 89, 65, 73, 81, 49, 52, 64, 76, 83, 92, 93, 68, 52, 79, 81, 83, 59, 82, 75, 82, 86, 90, 44, 62, 31, 36, 38, 42, 39, 83, 87, 56, 58, 23, 35, 76, 83, 85, 30, 68, 69, 83, 86, 43, 45, 39, 83, 75, 66, 83, 92, 75, 89, 66, 91, 27, 88, 89, 93, 42, 53, 69, 90, 55, 66, 49, 52, 83, 34, 36.

Now, we have 100 observations to present the data. In this case, we have more data when compared to example 1 and example 2. So, these data can be arranged in the tabular form called the grouped frequency table. Hence, we group the given data like 20-29, 30-39, 40-49, ….,90-99 (As our data is from 23 to 98). The grouping of data is called the “class interval” or “classes”, and the size of the class is called “class-size” or “class-width”.

In this case, the class size is 10. In each class, we have a lower-class limit and an upper-class limit. For example, if the class interval is 30-39, the lower-class limit is 30, and the upper-class limit is 39. Therefore, the least number in the class interval is called the lower-class limit and the greatest limit in the class interval is called upper-class limit.

Hence, the presentation of data in the grouped frequency table is given below:

Hence, the presentation of data in this form simplifies the data and it helps to enable the observer to understand the main feature of data at a glance.

Practice Problems

• The heights of 50 students (in cms) are given below. Present the data using the grouped frequency table by taking the class intervals as 160 -165, 165 -170, and so on.  Data: 161, 150, 154, 165, 168, 161, 154, 162, 150, 151, 162, 164, 171, 165, 158, 154, 156, 172, 160, 170, 153, 159, 161, 170, 162, 165, 166, 168, 165, 164, 154, 152, 153, 156, 158, 162, 160, 161, 173, 166, 161, 159, 162, 167, 168, 159, 158, 153, 154, 159.
• Three coins are tossed simultaneously and each time the number of heads occurring is noted and it is given below. Present the data using the frequency distribution table. Data: 0, 1, 2, 2, 1, 2, 3, 1, 3, 0, 1, 3, 1, 1, 2, 2, 0, 1, 2, 1, 3, 0, 0, 1, 1, 2, 3, 2, 2, 0.

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Definition of presentation of a group.

basing myself on suggestions I found in previous discussions, I have opted for the algebra book of Dummit. However, I have now a little problem. On page 26 (3th edition) the authors say that

"presentations" give an easy way of describing many groups, but there are a number of subtleties that need to be considered. One of this is that in an arbitrary presentation it may be difficult (or even impossible) to tell when two elements of the group (expressed in terms of the given generators) are equal. As a result it may not be evident what the order of the presented group is, or even whether the group is finite or infinite.

Then they report two examples

$\qquad$$\qquad$$\qquad$$\qquad$ $<x_1,y_1 \;|\; x_{1}^2=y_{1}^2=(x_{1}y_{1})^2=1>$

saying that one can show this is a presentation of a group of order 4, whereas

$\qquad$$\qquad$$\qquad$$\qquad$ $<x_2,y_2 \;|\; x_{2}^3=y_{2}^3=(x_{2}y_{2})^3=1>$

is a presentation of an infinite group.

The problems I encountered are essentially two:

1) I have found vague the "definition" of "relations"(and consequently that of "presentation"), because it is set metamathematicalwise (at least this way it sounds to me), using the concept of "derivation/deduction". Infact, no method is exhibited to clarify and rigorously formalize the expression used in the "definition", so I find some trouble, the underlying ideas having been left up in the air (what is meant, for example, as they say "it may be difficult (or even impossible) to tell when two elements of the group are equal"? what's the corresponding mental mechanism? and what if the $S$ generators set has an exorbitant cardinality? Is perhaps (also) this a motive of difficulty to which the authors refer?);

2) with reference to the two examples I mentioned, I am not able to understand the link between these and what has been said just before: there may be some problems about assigning the order to the presented group (it has been said), but the two presentations in the examples are well defined in this sense (I say).

• abstract-algebra
• group-theory
• group-presentation
• combinatorial-group-theory

• 1 $\begingroup$ The definition given here is intuitive. Formally we construct a group $G$ with set of generators $X$ and relations $R_i$ among the $X$ as a quotient of the free group on $X$. Specifically, if $F$ is a free group on $X$, then $G$ is defined to be $F/K$, where $K$ is the normal subgroup generated by the relations $R_i$. In the first example, $X = \{x_1, x_2\}$ and $R_1 = x_1^2$, $R_2 = y_1^2$, $R_3 = (x_1y_1)^2$ are the relations. $\endgroup$ –  spin Commented Apr 26, 2013 at 13:22
• 1 $\begingroup$ When giving a "relation" in a group presentation, one is asserting the given word is equal to the identity, e.g. $(x_1 y_1)^2 = 1$ in one of the examples you cite. Think of the relations as generating a normal subgroup of the free group on whatever generators are used. $\endgroup$ –  hardmath Commented Apr 26, 2013 at 13:23
• $\begingroup$ @spin yes, and thanks. But I ask if it has some utility present such an advanced topic at the very beginning... the reader (not possessing like me that notion), in my humble opinion, should be astonished at least... $\endgroup$ –  Bento Commented Apr 26, 2013 at 16:16

5 Answers 5

Let $\bar x$ be a vector of variables (e.g. $\bar x = a,b,c$).

Let $\bar R$ be a list of words formed from the variables in $\bar x$ (e.g. $aa$, $b^{-1}c^{-1}bc$).

The presentation $\langle \bar x \mid \bar R \rangle$ defines a group that is a quotient of [the free group on the symbols $\bar x$] by [the smallest relation on words of $\bar x$ that makes each word of $\bar R$ the identity].

A simple example is $\langle a \mid a^3 \rangle$, this defines the group $C_3$ because we start with the free group $\{\ldots, a^{-2}, a^{-1}, a, a^{2}, a^{3}, \ldots\}$ (this is isomorphic to $\mathbb Z$) and then quotient it out by the relation defined by $a^3 = 1$. That relation is the reflexive symmetric transitive closure of:

• $aaa \sim 1$
• $x^{-1} \sim y^{-1} \implies x \sim y$
• $x \sim x'$, $y \sim y' \implies xy \sim x'y'$

Throughout I used the term 'words', but it's really 'words plus inverses' but I don't have a name for that.

• $\begingroup$ thank you, I suppose this is the formal definition, but please consider my comment to Loki. Again, I will reread your answer afterwards. I will procede with order following the order of the book, without putting the cart before the horse. $\endgroup$ –  Bento Commented Apr 26, 2013 at 17:30

Here is a more abstract point of view.

The group $G=\langle x,y : x^2 = y^2 = (xy)^2 = 1 \rangle$ can be defined(!) by the universal property $\hom(G,H) \cong \{(a,b) \in H \times H : a^2 = b^2 = (ab)^2 = 1\}$, naturally in $H$, which is an arbitrary group. Thus, $G$ is the universal ("smallest") group containing two elements $x,y$ satisfying the three relations. One can easily check that $\mathbb{Z}/2 \times \mathbb{Z}/2$ solves this universal property, thus $G$ is isomorphic to this group. In particular, $G$ is finite, and abelian.

However, if $p$ is an odd prime, then $G=\langle x,y : x^p = y^p = (xy)^p = 1 \rangle$ is infinite and not abelian. This can be seen using the homomorphism $G \to \mathrm{Sym}(\mathbb{Z})$ corresponding to $a=(-p+1,-p+2,...,-1,0)(1,2,...,p)(p+1,...,2p)... $ and $b=...(-p+2,-p+3,...,0,1)(2,...,p+1)(p+2,...,2p+1)...$ (see MO/22459 ).

More generally, if $R_1(x_1,\dotsc,x_n),\dotsc,R_m(x_1,\dotsc,x_n)$ are group words in $x_1,\dotsc,x_n$, then

$$G=\langle x_1,\dotsc,x_n : R_1(x_1,\dotsc,x_n),\dotsc,R_m(x_1,\dotsc,x_n)=1 \rangle$$

is defined to be the group satisfying the universal property

$$\hom(G,H) \cong \{a \in H^n : R_1(a_1,\dotsc,a_n)=\dotsc=R_m(a_1,\dotsc,a_n)=1\}.$$

The uniqueness is justified by the Yoneda Lemma, and existence follows by taking $G=F_n/\langle \langle R_1,\dotsc,R_m \rangle \rangle$, where $F_n$ is the free group on $n$ generators and $\langle \langle - \rangle \rangle$ denotes the normal closure. In particular, the elements of $G$ are represented by group words in $x_1,\dotsc,x_n$, and we calculate modulo the relations $R_1,\dotsc,R_n$ and all relations derived inductively from them.

But in my opinion the universal property is far more important and also useful than the element description, which is even intractable in many interesting cases (see also hardmath's answer). For example, try to show that $\langle x,y : x^3 = y^3 = (xy)^3 = 1 \rangle$ is infinite with the element definition. Good luck!

You cannot really understand a group just by looking at its elements. You have to map the whole group it into more concrete groups. This is the whole point of representation theory, and actually is a special case of the philosophy of the Yoneda Lemma.

By the way, presentations are nothing special to groups. They apply to arbitrary algebraic structures. For example in the theory of $\mathbb{R}$-algebras, $\langle x : x^2=0 \rangle$ is the ring of dual numbers.

• $\begingroup$ some results you propose are not in my algebrical background, and I am at the present more interested in the question if the exposition is formally sensible. Also the connection of the two examples with the preceding assertions of the author, for which connection It seems to me that your answer gives indications. I think I'd better read your reply later (perhaps after finishing the group theory at least) for having plain consciousness of the matter. Eventually, why do you affirm the group to be abelian? I noticed (by an exercise) in the book that this is the dihedral, which is not abelian. $\endgroup$ –  Bento Commented Apr 26, 2013 at 17:03
• $\begingroup$ $D_2=\langle x,y : x^2=y^2=(xy)^2=1 \rangle$ is abelian, since it is generated by $x,y$ and $xyxy=1 \Rightarrow yx=x^{-1} y^{-1}=xy$. For $n>2$, the group $D_n = \langle x,y : x^n=y^2=(xy)^2=1 \rangle$ is not abelian. $\endgroup$ –  Martin Brandenburg Commented Apr 26, 2013 at 17:22
• 1 $\begingroup$ In order to understand my answer, no special group theory is necessary, only basic notions from category theory, namely the Yoneda Lemma and universal properties. For the deep understanding of algebra (and all related areas of mathematics) one needs to learn category theory. Even for basic notions in algebra, such as here the presentation of group, category theory helps a lot to organize the ideas. Feel free to ask questions! $\endgroup$ –  Martin Brandenburg Commented Apr 26, 2013 at 17:28
• $\begingroup$ Thank you very much! Casually I found this site, and I'm going to post questions anytime I need help:-) This is a good site, I have found a lot of competence, and helpful people. Sometime there's a little temptation of not thinking enough, since then someone will certainly give the solution, but in my little experience I try to get it by myself... at least until I must hoist the white flag. I choose the Dummit-Foote also because at the end of the book I noticed an Appendix about Category. (continue) $\endgroup$ –  Bento Commented Apr 26, 2013 at 18:55
• $\begingroup$ (continue) It was also vying Artin's book, in fact, reading comments on the site, it was even preferred by some, but I opted for DF because it contains more material. I have also learned that there is a classic of Birkhoff and Mac Lane that treats the matter from a point of view categorial. I've gave a look just now at the appendix and it seems to me, however, that there is not much (I do not see the lemma). $\endgroup$ –  Bento Commented Apr 26, 2013 at 19:03

What I know about the two groups is that they are from a family of groups called Von Dyke group $D(l,m,n)$. They can be presented as $$D(l,m,n)=\langle a,b\mid a^l=b^m=(ab)^n=1\rangle$$ Here, you see $m=n=l=2$ and $m=n=3$. You can google to find that this group is finite if and only if $$\frac{1}{l}+\frac{1}{m}+\frac{1}{n}>1$$

• $\begingroup$ this is interesting, and I think it's not present in the text (for the first one there's an exercise in which it is used the fact that the group presented is the dihedral, the second is just like an "ipse dixit"). $\endgroup$ –  Bento Commented Apr 26, 2013 at 16:23
• 1 $\begingroup$ You know your groups $\checkmark^{+1}$ $\endgroup$ –  amWhy Commented Apr 27, 2013 at 1:18

The difficulty or "impossibility" that Dummit refers to in your quoted material is the undecidability of the word problem for even finitely presented groups . Although for many group presentations the word problem can be solved algorithmically, a general solution was proved impossible by Pyotr Novikov in 1955 and again (using a different approach) by William Boone in 1958.

• $\begingroup$ I suspected something in such a sense. Thank you for replying, this make me more "relaxed" about the framework of the text... I think that I have just to wait for next chapters (well, I hope it will). $\endgroup$ –  Bento Commented Apr 26, 2013 at 16:03

Consider the free group on a set S of generators. From a set of simple statements $a \equiv b$ we can generate a congruence relation on the free group - an equivalence relation $\equiv$ such that $a \equiv b \implies ax \equiv bx, a/x \equiv b/x$ for all x in the free group. Congruence relations are homomorphisms with kernel $\{a|a \equiv 0\}$.

The laws the equivalence relation must satisfy depend on the signature of the algebra, and congruence relations can be used to define presentations for arbitrary varieties of algebra.

• $\begingroup$ what I said above I repeat it here. I am at the beginning of the text, I have the basics of abstract algebra, but not all there in the book (which is why I'm reading it). The free groups are among them, and I noticed that there are about at the end of group theory as well as the theory of representation (who I miss too). Thanks for the answer, I need it anyway to give me an idea, albeit approximate. $\endgroup$ –  Bento Commented Apr 26, 2013 at 17:18
• $\begingroup$ The free groups could just as easily be at the beginning of a group theory textbook. See if you can figure out what algebra has words made only from the symbols 0, 1 +, and parentheses, and where no words identify the same element if associativity and 0+x=x don't give you a rule for identifying them. $\endgroup$ –  Loki Clock Commented Apr 26, 2013 at 17:31
• $\begingroup$ I looked at the wiki-link hardmath gave, and I think I understand that, in summary, we proceed in this way: we consider the elements of a group and build a group formed by the expressions of the group (= sequences of elements) in relation to this group I suppose it happens the definition of presentation of the starting group. $\endgroup$ –  Bento Commented Apr 26, 2013 at 20:37
• $\begingroup$ But there are a lot of notions that are involved, so I remain with my opinion that it is premature to expose the concept just after the axioms of a group. (correction of the previous comment: ... AND in relation to this group I suppose it happens the definition of presentation of the starting group. ) $\endgroup$ –  Bento Commented Apr 26, 2013 at 20:43
• $\begingroup$ It's a lot less complicated than what hardmath is talking about. I think this is what you were saying about building a group formed by the expressions of the group: If you want $\Bbb{Z}_3$, you could present it as $<\{a,b\} | a+a=b, a+b=0>$ or as $<\{a\}|a+a+a=0>$. You could find out those expressions, $a+a=b$ and so on, from inspecting $\Bbb{Z}_3$. But say you don't apply any expressions, you just know there's this element $a$ and you assume it's in a group. That assumes it's closed, so $a+a$ is also in the group. You know $(a+a)+a=a+(a+a),$ because you assumed it's associative. What else? $\endgroup$ –  Loki Clock Commented Apr 26, 2013 at 21:00

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Mathematics is an endeavour of the human spirit

The science of pure mathematics, in its modern developments, may claim to be the most original creation of the human spiritFrancis Bacon

Axioms inMathematics

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Remember: Less is more.

A strong presentation is so much more than information pasted onto a series of slides with fancy backgrounds. Whether you’re pitching an idea, reporting market research, or sharing something else, a great presentation can give you a competitive advantage, and be a powerful tool when aiming to persuade, educate, or inspire others. Here are some unique elements that make a presentation stand out.

• Fonts: Sans Serif fonts such as Helvetica or Arial are preferred for their clean lines, which make them easy to digest at various sizes and distances. Limit the number of font styles to two: one for headings and another for body text, to avoid visual confusion or distractions.
• Colors: Colors can evoke emotions and highlight critical points, but their overuse can lead to a cluttered and confusing presentation. A limited palette of two to three main colors, complemented by a simple background, can help you draw attention to key elements without overwhelming the audience.
• Pictures: Pictures can communicate complex ideas quickly and memorably but choosing the right images is key. Images or pictures should be big (perhaps 20-25% of the page), bold, and have a clear purpose that complements the slide’s text.
• Layout: Don’t overcrowd your slides with too much information. When in doubt, adhere to the principle of simplicity, and aim for a clean and uncluttered layout with plenty of white space around text and images. Think phrases and bullets, not sentences.

As an intern or early career professional, chances are that you’ll be tasked with making or giving a presentation in the near future. Whether you’re pitching an idea, reporting market research, or sharing something else, a great presentation can give you a competitive advantage, and be a powerful tool when aiming to persuade, educate, or inspire others.

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Mathematics > Group Theory

Title: short presentations for transformation monoids.

Abstract: Due to the theorems of Cayley and Vagner-Preston, the full transformation monoids and the symmetric inverse monoids play analogous roles in the theory of monoids and inverse monoids, as the symmetric group does in the theory of groups. Every presentation for the finite full transformation monoids $T_n$, symmetric inverse monoids $I_n$, and partial transformation monoids $PT_n$ contains a monoid presentation for the symmetric group. In this paper we show that the number of relations required, in addition to those for the symmetric group, for each of these monoids are at least $4$, $3$, and $8$, respectively. We also give presentations for: $T_n$ with $4$ additional relations when $n$ is odd and $n\geq 5$; and $5$ additional relations for all $n\geq 4$; for $I_n$ with $3$ additional relations for all $n \geq 3$; and for $PT_n$ with $9$ relations for all $n\geq 4$. The presentations for $T_n$ and $I_n$ answer open problems in the literature.

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Feedback and assessment for presentations

Range of instructor feedback, specificity of instructor feedback, advantages of various forms of feedback, rubrics and grading/commenting forms.

Encourage students to improve their presentations: otherwise presenting repeatedly may merely ingrain bad habits. Feedback can come from peers and from instructors.

Consider commenting on the following:

• Timing notes: an outline of the talk including the amount of time spent on each portion.
• Feedback on the presentation style: style of speech, use of visual aids (blackboard/ slides/ images), pacing, audience engagement.
• Feedback on mathematical content: correctness, connections of material to other parts of course or other parts of mathematics (this is a good way to pique students’ interest in the subject matter).
• Feedback on teaching strategy: providing motivation, examples, conceptual explanations, repetition, etc.
• See also the general principles of communicating math .

Issues specific to various forms of presentations can be found on the page Assignments on Presentations .

The level of detail of the comments depends on whether the presentation will be given again. For example, noting every math mistake might be appropriate for a rehearsal so the student can be sure to fix those mistakes, but if the presentation will not be given again, a list of every mistake could be demoralizing with little positive benefit. At this point, comments should be more general and should focus instead on the sorts of things to consider for future presentations.

For other issues to consider when choosing and wording comments, see the handout Dimensions of Commenting .

• Most efficient is to take notes during the presentation and give them to the student immediately after the presentation.
• Most helpful for the student (but time intensive) may be to record the presentation and then sit with the student to review the recording.
• Another option is to discuss the presentation as a class immediately after the presentation. For this option to be successful, a respectful, collegial atmosphere is necessary.
• If you prefer time to think before giving feedback, you could e-mail your response after class or arrange to meet with the student at a later date. Meeting may be more efficient than e-mail because the student can ask clarifying questions so you don’t have to take the time to make your notes self-explanatory.

Identifying and prioritizing grading criteria before grading is important to prevent unintentional, subconscious bias,  even in graders who consider themselves objective,  as found by this study of hiring decisions based on criteria prioritized before/after learning about an applicant: Uhlmann and Cohen, “ Constructed Criteria: Redefining Merit to Justify Discrimination ,” Psychological Science, Vol 16, No 6, pp. 474-480, 2005.

Guidance for how to create a rubric is provided on the MAA Mathematical Communication page “ How can I objectively grade something as subjective as communication ?”

For classes in which each student gives multiple presentations, see the grading suggestions on the page for undergraduate seminars .

Sample grading criteria & rubrics for presentations are provided below.

Using a commenting form or grading form can remind you to consider all aspects of presentations that you’ve decided are important, rather than focusing only on the most obvious issues with any given presentation. A commenting form or grading form can also help you to find positive aspects of a presentation that on first consideration seems to be thoroughly troublesome. Some examples of forms and rubrics are below, but it’s best to make your own so the form reflects your priorities.

• Pedro Reis’ presentation evaluation form for M.I.T.’s Undergraduate Seminar in Physical Applied Mathematics , a topics seminar
• Characteristics of an effective undergraduate research talk : outlines basic expectations, characteristics of a good talk, and characteristics of an excellent talk
• Jardine, D. and Ferlini, V. “Assessing Student Oral Presentation of Mathematics,”   Supporting Assessment in Undergraduate Mathematics , The Mathematical Association of America, 2006, pp. 157-162 . This report of a department’s assessment of the teaching of math presentations contains a rubric for individual presentations. See Appendix B.
• Dennis, K. “Assessing Written and Oral Communication of Senior Projects,”  Supporting Assessment in Undergraduate Mathematics , The Mathematical Association of America, 2006, pp. 177-181 . Contains rubrics for presenting and writing, with recommendations.
• Rubric for Mathematical Presentations from Ball State University
• A description of criteria for math oral presentation for a math majors’ seminar, with categories Logic & Organization, Content, and Delivery.
• Form for commenting on and grading a presentation of a proof
• Scoring Rubric for Math Fair Projects with an audience of children
• Rubric for grades 6-8 for a math talk about solving two-step equations with one variable

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Microsoft 365 Life Hacks > Presentations > How many slides does your presentation need?

How many slides does your presentation need?

When you’re creating a presentation, it’s important to consider the amount of information you’re sharing with your audience. You don’t want to overwhelm them, but you also want to be comprehensive and ensure that you’re covering all your bases. Whether you’re giving a 10, 15, or 30-minute presentation, see how many slides your presentation needs to get your point across.

Rules and guidance for PowerPoint presentations

PowerPoint is a powerful visual aid for introducing data, statistics, and new concepts to any audience. In PowerPoint, you can create as many slides as you want—which might sound tempting at first. But length doesn’t always guarantee a successful presentation . Most presentations last around 10-15 minutes, and anything longer than that (such as a 30-minute presentation) may have additional visual aids or speakers to enhance your message.

A handy rule to keep in mind is to spend about 1-2 minutes on each slide. This will give you ample time to convey your message, let data sink in, and allow you to memorize your presentation . When you limit each slide to this length of time, you also need to be selective about how much information you put on each slide and avoid overloading your audience.

For 10-minute presentations

Ten minutes is usually considered the shortest amount of time you need for a successful presentation. For a shorter 10-minute presentation, you’ll need to be selective with your content. Limit your slide count to approximately 7 to 10 slides.

For 15-minute presentations

When preparing for a 15-minute presentation, concise and focused content is key. Aim for around 10 to 15 slides to maintain a good pace, which will fit with the 1-2 minute per slide rule.

For 30-minute presentations

A longer presentation gives you more room to delve deeper into your topic. But to maintain audience engagement, you’ll need to add interactivity , audience participation, and elements like animations . Aim for around 20 to 30 slides, allowing for a balanced distribution of content without overwhelming your audience.

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Using the 10-20-30 rule

The 10-20-30 rule is an effective way to structure your presentation. It calls for no more than 10 slides and no longer than 20 minutes (as well as a 30-point font).

Tips for crafting an effective presentation

No matter how long a presentation is, there are guidelines for crafting one to enhance understanding and retention. Keep these tips in mind when creating your PowerPoint masterpiece:

• Avoid overload: Ensure that each slide communicates a single idea clearly, avoiding cluttered layouts or excessive text.
• Pay attention to structure: Think of slides as bullet points with introductions, endings, and deep dives within each subject.
• Add visual appeal: Incorporate images, charts, and graphics to convey information without using too many words to make your audience read.
• Engage with your audience: Encourage interaction through questions, polls, or storytelling techniques to keep your audience actively involved.
• Put in the practice: Familiarize yourself with your slides and practice your delivery to refine your timing and confidence.

Ultimately, the ideal number of slides for your presentation depends on the allocated time frame and how detailed your content is. By striking a balance between informative content and engaging delivery, you can create a compelling presentation that can teach your audience something new.

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