What Doing My First (Short) Math Lecture Taught Me
For context, these are some things I learned in the process of putting together and delivering a guest lecture to a first-year discrete math course last summer. The talk was about the research I was doing at the time, and I was allotted about half an hour for the presentation. Again, I meant to write and post this last year, but clearly that didn’t happen. I am currently still catching up on my backlog of half written “reflection” type articles (which has apparently that’s taken me just over a year to start doing), so there are one or two more articles in the saga of research things I did last summer coming up.
1. If your talk is short, do not include a proof.
Even a sketch proof, unless it’s very very short, is probably too much for a half-hour or under presentation. If you do choose to include a proof, it will take up a lot of time and you risk losing most of the audience. (I stole this tip from my supervisor.)
2. Writing, speaking, and thinking simultaneously is hard, which makes delivering blackboard lectures especially challenging.
Blackboard talks involve simultaneously thinking about the content you’re delivering, the quality of your handwriting, how to speak, where to write things on the board, what to leave up and what to erase, and so on. Board work is very difficult, and writing while talking is also very hard. There is no way to do it well unless a few of those things have become automatic.
It’s also a very fluid kind of presentation, which is not ideal if you’re new to public speaking. You can never give the same blackboard talk twice, really: since you’re going to be writing everything up from scratch each time you do it and reacting to your audience in real time, things will change. You don’t have the crutch of resorting to reading slides, either, so if you screw up and freeze, everyone will notice; this is a medium that requires the presenter to be comfortable with improvising. This makes using a blackboard for your first ever presentation kinda dangerous, and I really don’t recommend it. (All of that time I spent doing algorithms assignments with my friends at various blackboards was necessary training for this, lol. So was all of my previous public speaking experience.)
3. Using a blackboard is just as much (or even more) work than using slides.
When using a blackboard, you still have to do the same amount of preparation as slides; however, what that preparation looks like is what will be different. Work out the examples you plan on using ahead of time and PRACTICE THE EXAMPLES; work out which parts you will actually write down on the board ahead of time and practice that as well; make some notes on index cards that you can reference during your talk (it’s way more convenient than having several loose leaf sheets of paper); and practice, practice, practice. There’s a physical aspect to doing a blackboard talk that can’t be emphasized enough: if you don’t practice it will look messy. (With slides, one can afford to wing it a lot more.)
4. Practice your presentation in the most similar room to your presentation venue that you can find.
The best way to practice this sort of thing is to give the lecture in a real room: your talk will change drastically once you attempt to actually do it. Practice your introduction, your conclusion, and every section of your talk. It might be worth trying a few different arrangements of your content, to see which ones feel better. I suggest practicing in empty classrooms or lecture halls at first, to have space to embarrass yourself in relative privacy while you workshop the first few iterations. Then, you should probably try timing yourself, taping yourself, and finally, in my supervisors’ word, “finding a victim” (a friend or colleague) or preferably, several “victims”, to listen to a few of your trial runs.
5. Practice your talk with real people, preferably people who aren’t super familiar with your work.
It’s best if not all of the people who help you practice are close collaborators on the work you’ll be presenting. You really don’t want them to risk mentally filling in gaps in your presentation. The sweet spot is people who are well-versed enough in your area to kinda know what you’re talking about, but who are new to the actual content of your presentation. That being said, people who have no background in your topic can still be effective audience members – they can tell you if your delivery is janky or engaging, whether the big picture organization of your presentation is coherent or not, and whether or not your explanations are confusing at a high level. (The best presentations can still kinda be followed by most viewers – a technical presentation is not an academic paper. Yes, it can be technical, but you also should spend some time talking about things at a high level. If your friends are more confused than seems normal, so will everyone else, most likely.)
6. Figure out how much your audience already knows about your topic.
You have to assume a baseline amount of knowledge on the part of your audience, and where your baseline is will significantly affect the kind of talk you give. You can’t possibly explain everything: for a beginner audience, you’re likely going to be explaining the basics in some amount of depth and glossing over some details (while still giving a glimpse of them). For a more advanced audience, you’re likely going to gloss over the basics and spend more time on the details (and probably still gloss over further details). It’s probably best to figure out roughly what kind of audience you have sooner rather than later.
7. Math is about communicating ideas; it is not about using fancy notation.
Communicating math is about explaining where the math comes from and making it tangible – draw pictures if you have to, and use examples if you have to. Math is about the ideas you communicate – notation is just a shortcut for communicating the ideas, and it takes a while for full absorption of the semantic meaning of notation to happen. In a talk, you don’t have that kind of time – you probably have about 15-30 minutes, so while you can and should use notation to explain yourself precisely, your audience won’t have time to absorb it if it’s new to them. What they will take away from your talk are the examples, pictures, and analogies you present them with, so make sure those are useful and reasonably accurate.
8. Storytelling and scaffolding are helpful tools for structuring a presentation.
Storytelling is really important in presentations, because it helps the audience follow along with your train of thought. It’s useful to use scaffolding (build a framework on which you can attach ideas throughout your talk) and analogies (people like when they have something solid to hold onto), and if you use an analogy at the beginning, it’s good to think about how you can keep it going and bring it around full circle throughout the talk. You probably want to start with the more concrete stuff and push the level of abstraction forward as you go, though in some cases the opposite approach may be better.
9. The order in which you introduce topics matters.
When I did my lecture on hypergraphs and the densest subgraph problem, I tried one version of my lecture in which I attempted to talk about approximation algorithms first, before ever introducing the concept of a graph or hypergraph. While technically, neither of these depend on the other, I realized that I would totally lose people if I talked about approximation algorithms first, because they’re a bit of an abstract concept and defining them involves quite a bit of notation that was new to my audience. In contrast, graphs have very visual representations, so kicking off the presentation with something concrete for my audience to look at was a much better move.
10. Get to the point: your presentation should have a single main idea.
30 minutes is not a very long time, and you will have to make some very difficult choices about what to include in your presentation. To be as effective as possible, you need to cut out all of the fluff and tangents. What is the core idea of your talk, and what do you actually need to get to the core idea? Even with this sort of narrow focus, you’ll probably still need to cut some things, so you’ll probably have to go through multiple iterations of deciding what your talk is actually about.
11. Make sure you actually know what you’re talking about.
Your presentation shouldn’t stretch right to the edges of your knowledge – keep it to the material you’re reasonably comfortable with. This is because doing a presentation requires you to know what the hell you’re talking about. First of all, if you’re up on stage, people are going to treat you like an authority on your topic, to a certain extent, so it’s best to avoid misleading people with your misconceptions. But, more importantly, you will probably need to answer questions from the audience, which will inevitably stretch outside of the material you explicitly presented. You need to know about more than just the talk contents, so that you can credibly answer questions. Yes, “I don’t know” is a valid answer, and one that I highly encourage using! (Please don’t make stuff up when you don’t know things.) But answering “I don’t know” to every question is embarrassing and not a good look.
12. Act like a confident person, even if you’re not feeling it.
Try to express outward confidence even if you’re freaking out inside, and be prepared to go with the flow a little bit. Things probably won’t 100% go as planned: that’s okay and normal. But once you’re on stage, your main goal should be to get through to the end of your talk, and it’s going to be much easier if you try your best to ignore any imposter feelings and focus 100% on giving the best delivery you can. While you’re presenting, the space is yours, so own it.