| Transcript
for the Math
Jam "HCSSiM Math Jam"
on Apr 1. |
| Math Jam hosted by smbelcas
(sarah-marie belcastro ). |
nsato19:29:32
Greetings and welcome to tonight's Hampshire College Summer Studies in Mathematics (HCSSiM) Math Jam.
nsato19:29:40
HCSSiM co-director sarah-marie belcastro is here to help students get an idea of what HCSSiM has to offer, and to answer questions regarding the program. I also understand that some HCSSiM alumni may be in attendance, and they can also share their experiences.
nsato19:29:49
But before I turn the floor over to her, I would like to briefly explain our classroom to students who have not previously attended a chat here.
nsato19:29:55
First of all, this classroom is moderated. This means that the messages you type will come to the moderators rather than going directly into the room. The moderators will choose some of the messages and questions to share with all of the students.
nsato19:30:02
Next, only the moderators have the ability to send private messages in the classroom. Sometimes they will use these to respond to your comments.
nsato19:30:08
At this time I will turn the floor over to sarah-marie.
smbelcas19:31:01
The Hampshire College Summer Studies in Mathematics is a 6-week residential program for highly talented high-school students. There's a ton of information at our website, http://www.hcssim.org but we're here tonight to offer personal perspectives on the program, in an interactive way.
smbelcas19:31:31
I am sarah-marie belcastro, Co-Director of HCSSiM. With me are HCSSiM alumns Hannah Alpert (first-year student at the University of Chicago and co-author on a significant research paper) and Sally Wolfe (homeschooled but takes college classes and has already participated in a Research Experience for Undergraduates). I like to brag about them and about many other HCSSiM alumns.
smbelcas19:32:07
(There are also other alumns in attendance, and I expect a few more to show up, too.)
snorkack19:32:10
I'm Hannah.
Ciaccona19:32:43
I'm Sally.
smbelcas19:33:37
Let me first ask: Any questions about HCSSiM? I also have some math up my sleeve, but I want to make sure that questions are answered.
patagonia19:34:09
what is it like?
smbelcas19:34:59
That's a sufficiently nonspecific question that I've no idea how to answer it. Alumns, your first/gut responses?
Bictor71719:36:19
Really good math and people. (Barry Weng '03 here)
snorkack19:38:13
I am tempted to make up some kind of nonsense metaphor like "It is like sitting in a hot tub full of brain-squishing octopi," but that would be foolish.
TimReyn19:38:13
you do lots of cool math that you never would have imagined would be cool, and then do a bunch of other stuff during the afternoons and weekends like seeing shakespeare, playing unexploded cow, playing frisbee, etc. (Tim Reynolds '07)
Rheingold19:38:29
You can really make it whatever you want it to be--there are so many people with different interests that depending on what you seek out, you'll have a different experience from other people--but above all, I think for everyone, you learn amazing things about math you never even knew existed, with equally amazing people (Megan Cutrofello, '07)
smbelcas19:39:11
Other questions? Or do you want some math?
patagonia19:41:32
How well do you have to do to get in and roughly how many apply each year?
smbelcas19:42:11
This all depends on the year. To get in, you must complete most of the Interesting Test correctly.
smbelcas19:42:31
In the past few years, we've had 145 - 160 applicants.
smbelcas19:42:51
This year, we have more than 102 already, and usually the bulk of our applications arrive in April and May.
TimReyn19:45:17
well i always want some math, that's a given
smbelcas19:45:27
Okay! Let's do some math. Here's a problem from the 2003 Interesting Test: The positive integers are colored yellow or pink in such a way that the sum of 2 differently colored numbers is yellow and their product is pink.
smbelcas19:45:55
What is the color of the product of 2 pink numbers?\r-and-\rCan you give examples of coloring satisfying this condition?
smbelcas19:46:16
Please toss out some thoughts and ideas.
philB19:47:01
yellow numbers are odd
even numbers are pink
apple pi19:47:01
It sounds like we can just have odd numbers = yellow, even = pink, right?
smbelcas19:47:17
That is indeed an example of a coloring satisfying the given condition.
smbelcas19:48:14
And that's a good first step in building intuition. But as apple pi says...
apple pi19:48:17
So if the color of the product of two pink numbers is uniquely determined, then obviously pink * pink = pink, but we don't know that there isn't a different coloring that also works
snorkack19:50:18
You can use the distributive property: pink * (yellow + pink) = pink * yellow + pink * pink.
smbelcas19:51:02
So, how does this finish the problem?
apple pi19:51:23
pink * yellow = pink + pink^2
apple pi19:51:35
pink = pink + pink^2
Rheingold19:52:52
What if pink^2 is yellow?
TimReyn19:53:12
so if pink^2 were yellow, then the right side would have to be yellow, but it's pink
apple pi19:53:16
then we have a contradiction!
apple pi19:53:21
because pink = pink + yellow = yellow
smbelcas19:53:31
Nice.
smbelcas19:53:37
This is a pretty slick solution.
smbelcas19:54:36
Another way of approaching the problem is to deduce what color 1 is.
TimReyn19:55:32
it has to be yellow, otherwise 1*yellow=pink*yellow=pink
smbelcas19:56:57
Now consider (1+pink) times pink.
smbelcas19:57:11
This leads to the same contradiction as earlier; it's not as slick.
smbelcas19:57:37
But it does show a way to deduce something about a number system when you're stuck:
smbelcas19:57:51
Consider the multiplicative identity.
isabella229619:58:12
what's that
smbelcas19:58:45
A multiplicative identity is something that doesn't affect what it is multiplied by, as in...
humblemastermind19:58:51
1*a=a
smbelcas19:59:19
But in systems other than the integers/reals, the multiplicative identity might have a name other than "1."
smbelcas20:00:02
If we were working on a problem that involved the additive identity (here we were just looking at the positive integers), we'd probably look at that too when stuck on a problem.
isabella229620:00:14
0?
smbelcas20:00:32
Yes, in the integers/reals 0 is the additive identity.
isabella229620:01:01
so pink^3 is yellow
smbelcas20:01:07
Please explain.
apple pi20:02:02
I don't think pink^3 can be yellow, because pink*pink = pink, and (pink*pink)*pink = pink*pink = pink.
smbelcas20:03:41
I think this last exchange is a pretty good example of how mathematics is done at HCSSiM---someone makes a conjecture, and then we investigate it. It doesn't matter whether the conjecture is correct or incorrect! What's interesting is what we learn from thinking about it.
smbelcas20:04:53
In general, Interesting Test problems are fairly loosely stated. Often there doesn't seem to be much information to use to work on them.
smbelcas20:05:09
But, there's usually a hidden hint in the statement of the problem. In this case, the hidden hint was the second question!
smbelcas20:07:29
This problem was pretty short compared to most Interesting Test problems, though. (It was actually one part of a three-part problem, where the three parts were not related other than by the idea of coloring numbers yellow and pink.)
smbelcas20:08:33
So! Back to questions about HCSSiM---please do ask some!
rponda200020:09:45
what ages is it for
smbelcas20:10:17
We don't have a set age range; most participants have just finished 10th or 11th grade.
smbelcas20:11:01
We always have some younger participants as well.
humblemastermind20:11:12
how much specific pre-knowledge do individual classes require?
smbelcas20:11:36
here we go... wait for it...
smbelcas20:11:40
...NONE!
rponda200020:11:54
12 years old?
smbelcas20:12:18
Rarely, but yes, sometimes. I believe that the youngest participant we've had in the last 36 years was 11 years old.
rponda200020:12:24
How do you qualify for it?
smbelcas20:12:57
There's an online application form at http://www.hcssim.org that asks for basic data like what classes have already been taken, name, address, etc.
smbelcas20:13:40
The form also asks for a sponsor name and a friendly letter explaining why you want to participate in the Summer Studies.
smbelcas20:13:45
And then, there's the Interesting Test.
snorkack20:13:50
It's unusual to participate in the program for more than one year, so it makes sense to go when you're older rather than very young.
Eagles20:13:56
How long do you work each day?
smbelcas20:14:04
You mean me, or the students? *grin*
smbelcas20:14:08
I work about 17 hours/day.
smbelcas20:14:24
The students do math for at least 7 hours/day, but usually more because they're having so much fun.
smbelcas20:14:40
(Alumns, feel free to chime in with comments, answers, etc.)
humblemastermind20:14:43
how large are the classes?
smbelcas20:15:36
During the first half of the program (3 weeks), classes are usually 12 - 15 students apiece with one senior faculty member (professor) and 1 or 2 junior faculty members (grad students or undergrads).
smbelcas20:16:01
During the second half, we also have mini-classes that are in the 4 -10 student range, and have a single faculty member.
snorkack20:16:27
Doing math for 7ish hours a day is very tiring the first week but gets easier. Don't be frightened because you get used to it if you don't usually do math for hours and hours at a time.
smbelcas20:17:51
Yes. There are also plenty of breaks.
snorkack20:18:30
During breaks you can eat but not juggle apples, if I remember correctly.
smbelcas20:19:11
I think that whether you can juggle apples during breaks or not depends on (a) whether you are indoors or outdoors and (b) whether anyone wants to eat the apples you intend to juggle.
Rheingold20:19:15
Also the math that you're doing is nothing like high school math in most cases--you go fast through really interesting stuff, so you don't get bored
smbelcas20:20:00
Let me add two things to that:
smbelcas20:21:16
(1) Frequently we present problems in silly ways. So for example, we might talk about returning wildly reproducing (and escaping) pets to the store, but secretly be discussing modular arithmetic. (Thanks to Cynthia Vinzant for that idea.)
smbelcas20:21:44
(2) Even the teachers find this stuff interesting and sometimes new---we teach material to each other as well as to the class!
Rheingold20:21:57
(I just realized that I made it sound like you don't go in-depth at all. that's really not true at all)
smbelcas20:22:41
Totally. The pace is fast, and we cover a lot of ground, and we go into depth. Sometimes the depth happens more on the problem sets (we have problem session in the evening) than in class.
smbelcas20:22:46
And problem session is really fun.
snorkack20:23:06
It's actually the depth that holds your (really my) attention. You get to figure out and prove the math yourself.
Ciaccona20:25:40
It's always so interesting or well motivated that it can be difficult to stop thinking about the math long enough to sleep and do other necessary things.
smbelcas20:26:12
Well, we do sometimes discuss math informally at meals.
smbelcas20:26:29
But I don't know any way of discussing math while asleep (unfortunately).
snorkack20:26:41
Sometimes the figuring-out of math gets really action-packed. Once a junior staff member (Emily Peters) had to tell some of us, "The point of HCSSiM is not to _yell_ about math!"
smbelcas20:27:21
...true, but _sometimes_ it's okay to yell about math.
smbelcas20:28:46
Other action-packed moments: a student charging across the top of a table in order to present a proof at the board... a student and faculty member faux-boxing while going back and forth about a possible hole in a proof...
smbelcas20:29:27
...students sneaking ninja-like to the board to add ideas...
smbelcas20:30:12
Y'all can ask more questions, y'know!
humblemastermind20:31:52
on the rare occasion you're not doing math, what do you do?
smbelcas20:32:03
Alumns, tell us!
snorkack20:32:09
Eat cheese. (really!)
smbelcas20:32:23
(We had a cheese-tasting "club" one year.)
apple pi20:32:28
frisbee?
smbelcas20:32:38
Yes. Some people play daily.
snorkack20:32:41
Dance.
TimReyn20:32:46
Card games (Fluxx! Bridge! Unexploded Cow!), board games (Settlers of Catan! Diplomacy! Monopoly! Risk!), frisbee, ......
Bictor71720:32:52
Set, Rubik's cube
Rheingold20:33:21
Lots of Set
snorkack20:33:23
There are also weekend activities like the hike and the trip to Boston.
TimReyn20:33:26
Work on the weekly program journal
Ciaccona20:33:28
Chamber music, juggling, and poi...
Bictor71720:34:09
Swimming, biking, nature walks
snorkack20:34:24
We had an MIT-inspired puzzle hunt in '05. My team won, ha!
Rheingold20:34:26
Going to a local produce market and eating fruit
smbelcas20:34:37
Sometimes people take naps, too.
Ciaccona20:35:43
Building mathematical objects (and very tall towers) out of Zome. There's a lot of origami too...
snorkack20:36:06
You can visit the National Yiddish Book Center or the Eric Carle museum on campus.
snorkack20:36:56
We also eat ice cream sometimes. Really yummy ice cream.
Rheingold20:37:00
There was a bocce tournament last year
apple pi20:38:47
bocce?
smbelcas20:39:14
It's a game played on a long clay pitch with a small target ball and sets of larger balls,
smbelcas20:40:04
colored to denote teams. (This is probably the worst description of bocce ever.) The goal is to have the largest number possible of your team's large balls be closer to the small target ball than any of the other teams' large balls.
Rheingold20:41:19
It's like curling, only with ground instead of ice and balls instead of irons
smbelcas20:41:38
Well, and no brooms.
smbelcas20:42:03
Other questions?
apple pi20:42:26
Are we going to discuss more IT problems?
smbelcas20:42:40
No, because we like to keep them under tight wraps.
smbelcas20:43:09
Plus, my pile of them is on a different floor of the house *grin*.
smbelcas20:44:51
And, many of them use diagrams, or require a lot of scratch work and so aren't suitable for an online classroom without lots of special preparation.
smbelcas20:45:23
It's not unusual for an IT problem to take more than an hour to complete.
smbelcas20:45:49
(Don't let that intimidate you, though! Every IT hour is a fun hour!)
snorkack20:47:03
But it's also possible to find the IT really hard and still turn out okay. Don't worry, you can always apply again another year (unless you get old).
smbelcas20:47:20
...where "old" means "graduate high school."
apple pi20:48:23
About how many students will be accepted this year?
smbelcas20:48:55
We will invite 40 - 55 students this year. (We accept all students as they are, but don't have room for everyone to participate.)
apple pi20:49:45
Do you try to avoid having too many people from one area?
smbelcas20:49:59
Yes, but that mainly applies to New York City.
smbelcas20:51:06
We do rolling invitations, so there is some advantage to completing one's application early.
smbelcas20:53:16
Well, it's been nearly an hour and a half, and we seem to be running out of questions...
smbelcas20:53:40
... do feel free to PM me, or to post a question or message on the HCSSiM forum..
smbelcas20:54:28
...and please be encouraged to apply to HCSSiM! The application is fun, free, obligationless (so you can and should apply to HCSSiM while applying to other programs as well), and doable online.
smbelcas20:55:07
Thanks for coming to the HCSSiM Math Jam!
