Just 5 minutes ago, I was taking a refreshing cold shower — because it’s too dang hot! And as best ideas are wont to come when paper and pencil are not around, I stumbled upon, in the rambling brambles of my nonlinear thought process, exactly what my summer project is going to be.
A little background first. The museum of math has had a series of math lectures (Math Encounters) this year, and so far I’ve been to all of them. They are delivered by people with grand speaking skills and on topics which are fascinating and excite the imagination. You can watch the first one (that they’ve put online) here:
I have decided I am going to try to come up with 3 lectures that I’m going to give in the first semester to students at my school. I’m thinking these are going to be after school things for anyone interested, and honestly, I will probably only get a couple teachers and a couple students to come. But it will be a fun lark for me this summer.
To be clear, I’m not talking about a workshop or problem solving sessions or anything super interactive. I’m actually thinking straight up lecture (with maybe some audience participation).
I’m excited enough about the idea that I think I will probably follow through on it. And so I thought I’d share the idea here, in case I can get others interested in doing the same. Anyone out there interested in doing it? I don’t think it would make sense for us to work together on the actual lectures, but I do think bandying about ideas for possible fun and high school accessible lecture topics could be superfun.
Just off the top of my head right now I have a few ideas: continued fractions, Farey sequences, the violent and sordid history of mathematics [I’d have to do some fun research on this one], topology, etc. Oooh! Coming up with good lecture titles will be EXTRA FUN!
My secret hope is that this is something that widens the scope of what kids think “math” is. I had that happen to me when I went to math camp in high school, where I was treated to so many amazing lectures on so many weird and fun topics that I saw the huge scope of math and saw the beauty even more piercingly than I had when exploring it on my own.
Continuous Everywhere but Differentiable Nowhere 
One of the things I wrote about in my PCMI application this year is that I really want to make the “higher level” math more accessable to high school students. Why do you have to wait until college to go topology or graph theory? I like the idea of the lecture with a bit of interaction, or at least places the audience could fill in the blanks. How long are you thinking? I know 10 minutes is about the cognitive limit for most people, but if the audience is there voluntarily, then I could see pushing that limit a bit.
Man, this is a really spiff idea. I look forward to working on it with you IN LESS THAN 23 DAYS! :D
Cool! I am definitely thinking of 40 or 60 minute lectures. Like real serious involved lectures for anyone who wants a little tidbit of this or that. The task will really be to come up with topics that are engaging and exciting, and find ways to bring them alive and asking for more. Or finding ways other people have done it, and cribbing them! I see no reason not to crib from others, if they’ve done a killer job, as long as you acknowledge you’re doing it to your audience (and send a nice thank you note to the person whose idea/activity/method of presentation you’re using)…
I do this in my classes! About once a month or so, we have a “blow your mind day” where I take time out of the curriculum to show the students “What Math is REALLY Like.” My recurring ones are: Different kinds of infinity (as seen on my blog), fibonacci sequence (and golden ratio), large numbers (how big is a trillion? What’s a googol?), is the world continuous or discrete debate (planck constant, atomic theory, is life a series of events or a continuous timeline, how we use discrete methods in calculus to approximate continuous math objects which approximate possibly discrete reality), knot theory (not much deeper than the Vi Hart level), among others.
It is fun! And also amazing (and sad?) how into it some kids can get as soon as you say, “This won’t be on a test.”
I second knot theory. Invented totally out of the joy of math, and decades later it is found useful in identifying protein strands in biology. Easy stuff to get started in to get the students involved, and it can get complicated fast.
I disagree with Dan about knot theory for proteins (though it is cool by itself).
One of the main people to study knots in proteins was Firas Khatib, one of my students, who looked at both knots and slipknots. The mathematics of knot theory has little to do with it, though, as proteins are not closed curves so knot polynomials (such as the Alexander polynomial) are totally the wrong way to look for knots in proteins. Some mathematicians had done that, and they got a lot of proteins wrong, since the knots they found were almost entirely formed by their artificial addition of a segment joining the endpoints of the protein backbone.
The best technique Firas found was a modification of one by Willie Taylor: the curves we are looking at are straight-line segments connecting a finite number of points, with the two endpoints fixed. If the triangle defined by 3 consecutive points has no other segments intersecting it, the curve can be simplified by removing the middle of the three points. If you can do this repeatedly until you have only a single segment joining the endpoints, then your initial curve has no knots.
We never proved the converse, that all knot-free curves can be simplified to a single segment by some order of point removal, and there are certainly orders for removing points that get you stuck with multiple segments in which all triangles have an intersecting edge. Luckily, using a couple of heuristic orderings (minimum area triangles first or shortest resulting segment after removing the intervening point) worked on all known protein structures to separate those with genuine knots from those that were knot-free. The number of proteins with genuine knots is very, very small, despite theoretical and empirical studies by physicists that just tumbling strings will usually form knots.
The concept Firas came up with for slipknots was more complicated to describe, so I’ll refer you to his PhD thesis:
Click to access Firas-Khatib-PhD-2008.pdf
Heh. I love the “sordid history” one — I would go to that. Have you read Two-Fisted Science by Jim Ottaviani? It’s a great series of graphic short-stories. The one about Newton and Leibniz getting into a bar brawl might be worth showing a panel or two.
Set Theory, group theory and maybe some symbolic logic. These can be taught intuitively (instead of axiomatically, which is probably too ambitious for high school). I did some (ultra) elementary group theory with some very bright 6th graders, and they loved it.
I like the idea of graph theory, both because it is useful for computer science and because discrete math gets no love in high school (and graph theory is way cooler than abstract algebra, which even has a boring name).
Fibonacci series and golden ratio is so overdone that I’d skip it.
I’m confused by CalcDave’s claim “how we use discrete methods in calculus to approximate continuous math objects”. Just the opposite is more common, with integrals used to approximate sums that are too difficult. Neither seems appropriate for a high-school fun extra.
Coordinate transformations (for computer games and perspective transformations) might be a good topic if you can make it accessible to math-phobic high schoolers.
Let me know when you have some talks – I’d love to invite you to present at the New York Math Circle’s Math and Dinner series. These are math talks to inspire math teachers, but we regularly attract others.
I’ll also be running a Bard Teachers’ Math Circle next year in the Bronx, and you’d be most welcome to come up and present there as well.
As a bonus, the bodega across the street from our Bronx location serves up a great Patacon, a sandwich with a fried and pressed plantain instead of bread. Mmmmm!
Mmmm patacon sounds incredible! And if I actually do it, and make something good, and muster up the courage, perhaps?
SAM
I have two suggestions:
1. serve food. you get more people and they are more prepared to learn with a few calories.
2. I LOVE combinatorial game theory. I think games like nim etc. are amazing in the way that they can be completely solved. Winning ways for your mathematical plays vol 1 is full of rigorous mathematics that is interesting and accessible.
I have done this kind of thing before, and I moving to the east coast (wallingford, ct). So if anyone is interested, let me know.
I was going to mention knot theory also–if nothing else there’s great visuals! You could also talk about hyperbolic, spherical and taxi-cab geometry. There are good textbooks out there on the history of math–the one I used covered all sorts of things including different cultures’ mathematics (ie, Egyptian symbols, different base systems, etc)
Right on! I like the idea of doing lectures as an educational supplement and not as an instructional method. It’s your opportunity to profess … and not have to worry about the consequences on the listening side. My advice is to go deep.
On the topic of interesting history, mathoverflow http://mathoverflow.net/questions/53122/mathematical-urban-legends/53617#53617 recounts a tale of Turan where Number Theory saves his skin. Plenty more where that came from.
Brainstorming the violent/sordid –
* I know you’ll mention Galois.
* The story of the general solution to the cubic in the early 1500’s is mega-dramatic, in a reality tv way. Look up biographies for Cardano, Tartaglia, and Ferrari. Also, Cardano’s life apart from the solution to the cubic also had a lot of violent and sordid.
* The story about Archimedes being killed by the invading soldier b/c he wouldn’t look up from his problem is probably not historically verifiable, but supposedly it was the inspiration for Sophie Germain’s career as a mathematician? (Look this up?)
* Oh snap, def. look up the biography of ALAN TURING: espionage and code cracking, life of a gay man in a homophobic society, with corresponding tales of illicit sex, unjust prosecution and creepy coerced hormone treatment regimes, etc… Turing is one of my favorite mathematicians-in-history and I would love for his story to be better known…
I have a couple of ideas of what you could talk about, as I did something like this a couple of years ago: Discrete Math for the High School Classroom, Part 1 and Part 2.
I would caution you to not design these as pure lectures. There’s so much you need to *show* to really do these types of higher level concepts justice.