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.