Classical (Oral) Comps
Obtaining your comps topic
Please fill out the following questionnaire to help the mathematics faculty in choosing your comps topic. Turn in the completed questionnaire to Sue Jandro.
Note: if you wish to receive your topic during a break, you must submit this questionnaire to Sue Jandro at least 3 weeks before the end of the term (the term preceding the one during which you wish to give your talk).
- Reading technical material
- Preparing your talk
- Giving your talk
- What happens at the two talks?
The following is a compilation of suggestions by various authors on the art of reading and presenting technical material. These suggestions are meant as a resource for students who would like advice on how to begin preparing for the oral comps in mathematics.
The topic you have been given for your oral comprehensive exam may be a general area or it may be as specific as a particular theorem or algorithm. In either case, you will want to go to the library and search for texts and journals that contain articles related to it. A source that keeps details to a minimum, such as a chapter in a general book giving an overview of your topic, is a helpful first step. You should also check the Web, especially if your subject is new enough to be changing rapidly.
After you have found one or several sources on your topic, you must work your way through them until you have a sense of what information they hold. Keep in mind that the first few sources you find may not be the ones that will prove most helpful to you.
Reading technical material in an area unfamiliar to you is much like reading an article in a foreign language; you may see many phrases and symbols that you haven't encountered before. Here are some suggestions to help you attack the pages of reading ahead of you.
At first, you probably won't be able to read everything in detail. Don't try. Instead, you should plan on making several passes over the literature, gaining more information on each pass. Keep paper and pen with you as you read to make notes. (Some people prefer to take notes on a computer. Do whatever is best for you, but don't forget that the combination of paper and pen is a popular technology with excellent features, including reliability, portability, and flexibility.)
On the first pass through, you should learn something about your topic:
- What's the point of the article?
- What problem is being addressed?
- Why is the problem interesting?
- What sub-disciplines of mathematics, statistics or computer science are used?
- What is the history of this topic?
- What are the results? (Results can include theorems, algorithms, etc.)
- Are these results simple or difficult?
- Should this source be part of your presentation or not?
You should not spend very long on the first pass through the material. Get in, answer the questions, and get out. Don't let yourself get bogged down trying to understand symbols and terms that may not be relevant to your talk.
After you have made a first, brief pass through all the material, make a preliminary outline of what you want to cover in your talk. (See the section on Preparing Your Talk for advice on what to cover.) Then return to the relevant sections of the articles, and read them in more detail.
On the next pass through the material:
- Get the big picture: read to understand how results, techniques, data structures, etc., fit in context with each other.
- Make a list of terms and symbols you will need to look up to understand the material.
- Again, avoid going through the material line by line. At this stage don't worry about filling in all the details of proofs, or every step of each algorithm. It might be helpful to prove for yourself some things labeled "obvious" or "trivial" to find out if you're understanding.
After you have the big picture of what you would like to present, you are ready for your third (but probably not final) pass through the information. This time, you should:
- Read proofs in detail. It is a good idea to make proofs your own by first thinking about how you would attack the problem before you read the details of the proof presented in the article. Outlining a proof on your own paper as you work through it is a good way to help you see the big picture of the proof.
- Try small examples. This can help for proofs and algorithms. For example, you might follow the steps of a proof of a theorem about groups while imagining the group is cyclic group, or the integers under addition. Or see how a graph planarity testing algorithm determines that the complete graph on five vertices isn't planar. Look up unfamiliar terms or notation. If it's unfamiliar to you, it is almost certain to be unfamiliar to your audience as well.
Once you have read all your sources and you have some ideas about your topic, you are ready to sit down and work on preparing your talk.
You have a great deal of freedom to pick the approach you use; your committee will not have very definite expectations on how you should approach your topic. However, there are some pointers to keep in mind for a good talk.
A well-constructed talk has a beginning, a middle, and an end, with a clear focus throughout. (Some say "tell them what you're going to say, say it, and then tell them what you've said.")
In the beginning, give the audience some background information about your topic, possibly giving a brief history, or setting the topic into context. This is to be a comprehensive exam: you should demonstrate that you can grasp the big picture and how your topic fits into that big picture. The start of your talk is also a good time to give motivation as to why your topic is interesting, such as real-world applications or relationships to other well-known problems or importance in its own field. If you will be doing some complicated mathematics, a simplified, easily-grasped example of what you will be showing or proving will engage your audience. Another approach is to pose an interesting question that leads the audience into your talk and which you answer (or perhaps prove cannot be answered) during your presentation.
In the main section of your talk, you will present what you find to be the most interesting or most important aspects of the topic you were given. You may choose to explain an algorithm or prove a theorem in detail, or show several different approaches to your topic, or pursue one line of thought in depth. Whatever you choose, take particular care to reinforce the direction you are going and where you've been, since your listeners won't have time to pause and review what has gone before, and certainly can't scan ahead to see what is coming.
The last few minutes of your talk could be a short summary of the whole lecture, or a placement of your topic back into its context, or a statement of the most important unsolved problem in the subject, or something else. Whatever you decide it should be, your audience and your talk deserve a conclusion.
Other things to keep in mind include:
- Talk about what you understand. If you don't understand it, the audience won't either.
- Keep jargon and symbols to an absolute minimum. Only introduce terms or definitions that you actually use. Remind the audience of the meaning of a new term the first time you use it. If you will be using particular terms or examples frequently, arrange to leave the important information in a place where your audience can see it throughout your talk (for example, on a side blackboard, an extra overhead projector, or a handout).
- Give a careful derivation or proof if it will help the audience understand or appreciate a result. Remember that some details do not need to be worked out completely--explaining briefly that an equation can be solved by elementary algebra can be more helpful than showing the solution step-by-step.
- Use examples and/or pictures and/or models and/or demonstrations! To do what?
- Motivate a part of your talk,
- Clarify a difficult definition,
- Illustrate an algorithm,
- Demonstrate the complexity differences between algorithms,
- Demonstrate the need for certain hypotheses of a theorem,
- and so on.
Abstract ideas are almost always easier to understand when they are made concrete.
- Vary your method of presentation. If you want the audience to follow a derivation or proof in detail, you should consider doing it on the blackboard, since writing slows you down so the audience can follow you. You can do drawings or proofs or an outline of the talk on transparencies (also called slides) or by using presentation software such as PowerPoint. Physical models can be passed around. This talk is unique to you and carries your own personal stamp.
- Prepare some "extra" material near the end of your talk--another example of some theorem or illustration of the applications of a program to another science--that can be added if your talk appears to be running short, or jettisoned if you appear to be running over time. Make it something that the audience will never know was added or removed. In particular, don't make future computations or examples depend upon it. Then as you practice your talk, get to know where you should be after 15 minutes, 30 minutes, and 45 minutes. On the day of your final presentation, you'll know in a glance to the clock if the material is needed or not. (Be careful not to watch the clock during your own talk; it's distracting to the audience.)
- Both students and faculty will attend your talk, so you should begin at a level that your peers will understand, but by the end you should be addressing (or even challenging) the faculty present. How you make this transition will depend somewhat on the topic you were given and the strength of your own background. There are competing goals for this comprehensive exam: you want to help your audience understand the material and you want to show the faculty that you've challenged yourself and pushed your own horizons. Keep in mind your audience as you are giving definitions, applying theorems, and giving examples.
- Write ideas down for topics, algorithms, theorems, examples, proofs, or historical background that you think you would like to include. Don't scrutinize them as you write them down.
- Organize your ideas into an outline for your talk. Spending time preparing an outline will make actually writing the talk much easier.
- Write out your talk, including exactly what to put on the blackboard (with a clear idea of when it will be put there and whether it will remain for long or be rubbed out right away), and what to write on slides. Work out the slides roughly on paper first. Write very large (see the section on making slides), and pare down what you plan to put on the slides. These talk notes should help you to be comfortable with the material, and are not intended for you to read to your audience during your presentation.
- For your own benefit, prepare an abstract for your talk. That is, try to summarize in a few sentences what it is you'd like to convey in your hour.
- Again think about what you want to include, how you should organize your ideas, and abstract your talk. Be willing to make changes to your original plans!
- Plan what you will say between (and during) slides and board work.
- Practice and time the talk. Keep in mind that times may vary depending on how relaxed you are, whether your audience asks questions, and so on. Revise the talk for length: don't talk faster, make the talk shorter!
- Practice the talk several more times; feel free to practice in front of a friend and get comments on your presentation technique.
- Revise the talk once more. Make final versions of your slides and rewrite your notes so that you can read them easily. Write down reminders to yourself (such as "pause" or "slow down here"). Number your notes and transparencies so you can reorganize them if you drop them.
- Practice the talk one last time. A good trick for overcoming stage fright is to memorize your first sentence, after which your practice and knowledge will take over.
- Ask yourself again what the big picture is, and try to anticipate questions that you may be asked during and after your talk and how you would answer them.
- You're ready!
Here are a few things for you to consider when preparing and rehearsing your talk.
- Arrive at least ten minutes ahead of time to get set up. If you plan to use a computer, you should warn Mike Tie at least three days in advance, and then show up for your talk twenty or thirty minutes early to make sure the hardware and software are ready to go.
- Set a time goal of 60 minutes for your delivery, including 5 minutes at the end for questions. Expect questions during the talk.
- A few moments of silence (five seconds, say), strategically placed after the statement of an important theorem or difficult definition puts the audience on notice that ``this is important" and gives them an opportunity to absorb what you said.
- Speak slowly and speak loudly. Be certain the people at the back of the room can hear you.
- Speak to the people, not to the board or overhead projector.
- Make eye contact with the audience members.
- Although it's fine to bring an outline of your talk on paper or on notecards (and perhaps a detailed definition or example written out completely), you should not be wholly dependent on your notes or notecards for the talk.
- Be enthusiastic about what you have to say.
- Get rid of potential distractions such as keys, loose change, or cell phones.
- Bring some water or tissues to your talk if you might need them.
- Take a few deep breaths and relax. You have prepared well and you are an expert on your topic. Your audience is rooting for you.
Slides for an overhead projector can be obtained from Sue Jandro in the Math/CS office. You can write on them with multi-colored pens (permanent or "spit-erase") or transfer figures or typeset pages directly using a copy machine. If you are familiar with TEX or LATEX, you might find SliTEX useful as you prepare your slides. Color-coding various aspects of your talk may be helpful, though changing colors randomly or using too many colors can be distracting.
An important rule of slides is: less is more. The most common mistake in writing a talk is to put too much on the individual slides. A busy, cluttered slide is hard for the audience to assimilate and may divert their attention from what you are saying. For the greatest impact, use relatively few slides, with simple, carefully chosen images. Everything on a slide should be essential; if not, get rid of it. Put only what is logically necessary on a single slide, not what is physically possible.
Write your characters so that they can be read easily from the back of the room. Leave at least a one inch margin top and bottom to avoid having parts cut off. A good rule of thumb is to have 10 or fewer lines per slide. If you have figures or pictures, make sure that the images are visible from far away. If not, enlarge them on a copy machine.
After you put your slide down, step away from the projector so the audience can see it. Look at the screen to check the picture. To draw attention to part of a slide, point to the screen with a long pointer or pen (with the cap on!). You could also lay a pen down directly on the slide or have several arrows cut out of paper that you can lay on the slide pointing in the right direction. Hands tend to shake, so avoid using your finger as a pointer. Holding onto the slide while talking makes it wiggle, so let go and step away.
Leave slides up as long as possible; many people read slowly. Aim for economy of words as you write a slide. As a visual aid, a slide should contain the core of what you want to say, but you can fill in the details and explanation as you talk through your slide. (If you merely read the slide, it could be argued that you might as well not be there!)
You should number your slides in case they get dropped, and place pieces of paper separating the slides. These not only help protect the slides, but they can contain key words, phrases, ideas, or encouragement that you need to remember at that particular point of your talk. You can read the notes as you're putting the slides on the projector, and no one is the wiser.
If a development stretches over several slides, the audience may wish to refer back to an earlier slide. To prevent this, you can replicate information from one slide to another. Some use a related technique of covering a complete slide with a sheet of paper, and gradually revealing the contents; but be warned that some people find this peek-a-boo style irritating. If you choose this approach, cover the slide before it goes on the projector, not after. If you think you will need to refer back to an earlier slide at some particular point, insert a duplicate slide. This avoids the need to search through the pile of used slides. Also, you can arrange in advance to have two overhead projectors for your use.
You can overlay slides to show a construction or algorithm in stages. If you overlay one or more transparencies, it is a good idea to tape them together into a "book"; it's hard to line up figures when you're nervous.
If you have never used a chalkboard before, or have a difficult figure to draw, practice in advance. Write large and press hard.
Erase the entire board before you begin. Strategically place chalk and erasers where they will be needed. If you want to use colored chalk, bring your own. If the chalk squeaks or vibrates, break it in half or hold it at 45 degree angle to the board.
Use the boards from left to right. Avoid writing much prose on the board (usually no one is taking notes). Intelligently chosen abbreviations, used consistently, are encouraged.
If you have an enlightening picture or example to talk through, the audience will delight in seeing the visual presentation grow before its eyes. Don't cheat them out of this display by having it drawn on the board in advance. However, for very difficult pictures, or those where precision is important, arriving before the talk and preparing the blackboard (or a slide) may be appropriate.
If you want to give an argument step-by-step, a chalkboard is useful, since you can refer back to definitions or earlier steps. Also, writing on the board can help you slow your pace.
Give the audience as much time as possible to read and absorb what is on the board. Don't erase anything until absolutely necessary. In particular, it is not necessary to erase the board after you've finished a subtopic; instead allow your audience the additional time to soak it in until you actually need the board. After you write something down, step away from the board so that everyone can see. Avoid talking while facing the board. Repeat what you have written after it is complete.
More and more students are using presentation software or Web browsers to organize and present their talks. This is great--demos can make a subject come alive, and well organized hypertext-based visual aids are easy to navigate and can be posted on the Web for anyone who misses your talk.
Most of the advice given above for using slides applies to computer presentations as well: use big fonts (18 and 24-point usually work best), don't pack too much information on a single page, leave the information visible long enough for your audience to read it, and so on. In addition, you should build in links that make it easy for you to navigate both forward and backward between pages, since you might wish to refer back to an earlier page, or compare two pages side-by-side.
If you use a computer, there is a chance that it won't work, leaving you with no talk at all. To reduce the probability of such a catastrophe, you should enlist Mike Tie's help early (a week in advance is good), practice your talk a couple days ahead of time in the room where you will give it, and show up for your talk early enough to make sure the computer is working properly.
All the comments above apply to your first talk. Your committee expects you to have prepared this talk as if it were your public talk. That is, the first talk is considered a dress rehearsal, not a first draft. You should prepare your slides and rehearse for this penultimate version. In addition to slides, notes, or models that you need to bring to this talk, bring along a pad of paper and a pen. After the talk is finished, the faculty present may ask you questions which you will want to investigate, and they will give you advice on how to improve your talk in the following two weeks. During those two weeks you may speak to faculty members if you have additional questions.
The only difference between the first talk and the final talk for many of you is that you'll want to address the issues raised by the faculty at the end of your first talk. Heed the advice you receive, it comes from well-meaning, seasoned professionals. Make use of the time before your final talk to ask questions of faculty about the particulars of any material you were uncertain about during the first talk and to practice your talk again and become even more comfortable with the material. (At this stage, feel free to invite other majors to hear and comment on your practice talks.) Now that you have an overview of the subject, you might want to step back from it and try to view it as an audience member would. What questions come to mind as you see it in this new light? How would you answer those questions if they occur at your final talk?
Immediately after your first talk, schedule the time and place of your second talk with Sue Jandro so a room can be reserved and the talk advertised on the department's Web page calendar. Also, make sure to give the Goodsell Gazette editor a title and abstract for your final talk well before the talk is scheduled to be given. These should keep technical terms to a minimum and be interesting enough to entice your colleagues to attend.
Within a week or so after your final talk you will receive a note from the department giving its evaluation of your performance on this segment of your comprehensive exam.
The department faculty wish you the best and want you to succeed. Show everyone (including yourself) what a professional job you can do. Good Luck!
- Deborah S. Franzblau, PRIMUS, II (March 1992), pp. 16-32.
- P.R. Halmos, Notices of the AMS, 21 (April 1974), pp. 155-158.
- Nicholas J. Higham, Handbook of Writing for the Mathematical Sciences, SIAM, Philadelphia, Pennsylvania, 1993. ISBN 0-89871-314-5.
- Large portions of this are excerpted from: Franzblau , Halmos , Higham . Compiled by Deanna Haunsperger; edited by Jeff Ondich.