There are many useful websites containing advice on how to give a good mathematics presentation (such as those listed here). But these are written for scholars who are giving lectures on their research. They focus on organizing the talk, putting the research in context, deciding what to include or not include, designing slides, pacing and time management, pitching the talk to the right level, and so on. This is all useful advice, but it is not necessarily relevant for students wanting to give a high-quality presentation of a proof or a problem to their class. What follows is advice specifically written for those students.

First things first: Everyone gets nervous giving a public presentation. Throw in a tricky math problem or a proof on a topic they have just learned, and it can be a source of great anxiety. It is natural and okay to be nervous. Your professors get nervous too! The best way to calm your nerves is to be well prepared before you step in front of your classmates. Here are some specific things you should do before you set foot in class.

• All ducks in a row. Control what you can control, and leave as little to chance as possible. Walk into class as prepared as you can be.
• Do the math. Of course, your first task is to solve the problem or prove the theorem. If you are confident with your work, you will feel less anxiety when you present it. If you are unsure of your solution, go through the mathematics with your professor or a classmate beforehand.
• Dress rehearsal. Give a sample presentation. You can do this by yourself, or you can present it to a friend, your professor, or your cat or dog. Reading over your proof ahead of time is not the same as giving a presentation. You will often find flaws, gaps, or tricky spots that need elaboration when you present your work. It is better that this happens during a trial run than during the actual presentation.
• Oops! Don’t worry about making mistakes. Everyone makes mistakes. After your presentation, the professor and the students will give you feedback on your work. Take these comments as suggestions for improvement rather than attacks on your work.
• Know it, don’t memorize it. Do not try to memorize your proof or solution. You should be well-practiced and familiar enough with the mathematics that you can present it naturally.
• Don’t “wing it.” Walking into the class you may think you can prove the theorem or solve the problem without using notes or without being fully prepared. However, mental computations and logical thinking may be trickier to carry out when you are nervous. You may have a mental lapse and forget key details when you are standing in front of the class. It happens to us all! In such a situation, it is nice to have a safety net—make sure your notes are clear enough that they can help you string the argument together. Lastly, don’t be spontaneous; you may regret trying a new approach to the problem while you are standing at the board.

Now that you are prepared, you are ready to present the mathematics to the class. Here are some comments and suggestions for how best to deliver the material.

• What’s the point? The main purpose of your presentation is to communicate mathematical ideas to your audience. You are not simply “putting your proof on the board” or demonstrating to your professor that you completed the assignment. You are not trying to convince your classmates that you are smart or clever. You are there to teach them some mathematics.
• Writing is important. Keep in mind all of the mathematical proof-writing advice you have learned, and write a high-quality proof on the board. If you are solving a problem instead of writing a proof, you may be able to be a little more sketchy and informal. However, keep in mind that your classmates are taking notes, and what you write on the board is likely going to be exactly what ends up in their notes. The more clear you are, the better their notes will be.
• Give the big picture. We all know from learning new mathematics that it can be difficult to see the forest for the trees. Your job in presenting your work to the class is to help them understand the big picture. For instance, rather than simply working through a set theory proof one line at a time, you can start by saying something like, “The key to this proof is showing that the set A has one and only one element.”
• Remember your audience. Pitch the proof and the discussion to your audience. Do not over-explain elementary ideas that your audience can grasp easily. Also, do not breeze over complicated or technical ideas. You can often read your audience by observing their body language. But if you are unsure whether they are following a particular argument, ask them.
• Let me give you a minute. Keep in mind that your audience will be taking notes, and they will be trying to follow the logical progression of the argument. It is difficult for them to listen and write at the same time. Don’t rush through a proof so fast that they cannot keep up. Pause after each sentence and visually check in with the class. If it looks like they are confused or if they are trying to get your attention to alert you to an error, address the issue.
• Notes or no notes? An ideal situation is to present the proof without a glance at your notes. However, this is often unrealistic and unnecessary. It is acceptable and appropriate to have a page or two of notes at hand so you can check them from time to time (do not bring your entire binder with you). Notes can help you remember what the next step is, they may contain some key wording that you need to ensure is precise, and they can give you a way to check that you haven’t omitted a key detail. However, you should never simply copy work from your notes.
• All the world’s a stage. Although you are not performing as an actor, you are performing. Be conscious of the way you deliver the content. Speak clearly, loudly, and slowly. Be enthusiastic, smile, find the right pacing, and connect with the audience. Make eye contact. Move from person to person. (It would be awkward to make eye contact with the same person the entire time—including your professor!) Do not stare up at the ceiling, down at the floor, at your notes, or at the board. Be aware of your nervous actions—”ummm”s, “ahhh”s, “y’know”s, “like”s, fidgeting hand gestures, and so on.
• Writing and speaking. Find a good balance between writing, speaking, facing the room, and facing the board. If you can write and speak at the same time, great. But any time you are not writing, you should face the class. Don’t write multiple sentences in silence. Do not speak into the board unless you are speaking and writing at the same time.
• Font size. Your writing should be large enough and neat enough that it is legible at the back of the room. Print; do not write in cursive. Use a marker color dark enough that it is easy to see. Write in horizontal lines—your sentences should not slant up or down.
• Move it. Walk around if you can. Most importantly, do not stand in front of your work when you are not writing. Give the entire class a full view of your work.
• Wipe out. Erase enough of the board to have a nice, clear area to write on. Do not squeeze your work in and around other writing that is already on the board. However, do not erase the existing work until everyone has copied it. When you do erase, use an eraser, not your hands. If you do use your hands, don’t then touch your face or your clothes as it may leave a colored smudge behind. Pro tip: erase the board up-and-down, not side-to-side. If your erasing arm moves side-to-side, then, by conservation of momentum, your torso will dance back-and-forth to compensate.
• Use your arm. Writing on a chalkboard or a whiteboard is different than writing on paper. Do not rest your palm on the board, and write using your hand muscles like you would with a pencil. The only thing touching the board should be the tip of the chalk or marker; use your arm to write.
• A thousand words. Many proofs or problems have a picture that must accompany the mathematics. Obviously, in those cases, you should draw and reference the figure. Be sure it is large enough and all the key features are clearly labeled. You can use colored markers to highlight certain aspects, but keep in mind that your note-taking classmates may not have multiple colors at their disposal. There are many cases in which the theorem or problem is abstract and there is no exact figure that accompanies it. But it still may be helpful to draw a representative figure that illustrates the ideas you are presenting.
• Know when to say when. If your chalk squeaks, break it in half. If your whiteboard marker is out of ink, throw it away.
• A topologist, an algebraist, and an analyst walk into a bar. Be careful about telling jokes. A class is typically not the appropriate venue for jokes, jokes often fall flat, and an inappropriate joke or one taken the wrong way can create new problems. Also, even if you are nervous, avoid self-deprecating jokes.
• No negative talk. You are the expert when presenting a problem to the class. Proceed with confidence and do not talk badly about yourself or your abilities. (“I’ll probably get this wrong.”) However, if there is a part of the proof or problem that you are unsure about, be honest about that. It would be best to take care of the problem before class, but if it arises while you are presenting, ask the other students or your professor for help. (“I’m not sure if I can say this. Do I have to justify it further?”)

[Image credit: fauxels]