A little backstory:

I do all my work in Algebra II on a SmartBoard. I always post PDFs of each day’s notes for students to download if they want to refer to something. A few times a week, a student will ask me — when I’m moving to a new page of the SmartBoard — to go back so they could finish writing their notes. I’d say almost every time I go back and let them finish. But sometimes I have to move on in the lesson, and the student asking decided to write things down after the material has been on the screen for a few minutes already — right when we’re done with that material. In those cases, I say “I’m sorry I’m going to go forward, but remember, I always post the SmartBoard.”

Now the story.

Yesterday I was in Algebra II and we were working on optimization word problems (find the maximum volume of box with some constraints on the sides, find the minimum cost to make a box). I realized I had been up at the SmartBoard too much recently, so I called on a student to be “Teacher.” I told the student that she had total control — so she needed to ask questions, answer questions, and keep all our miscreant compatriots in order.

She started working, and doing a fabulous job. She fielded questions like a champ. At one point, she moved from one page to the next, and someone asked if she could go back to the previous page.

She looked at me, at the student, at me again, got the smallest smile on her face and said:

“I’m sorry I am moving on. But I post the SmartBoard every night.”

The whole class broke into laughter. I laughed. And laughed. I couldn’t stop laughing for like an entire 60 seconds. Classic.

I like that class a whole lot.

(And indeed, after the laughter subsided she didn’t go back to the previous SmartBoard page, but kept on working.)

I have been in a bit of a nostalgic mood tonight, and so I went back to look at the journal I kept in college. In it, I found an entry where I feel statistics was used improperly. Not by me, but by my Algebra professor.

here’s the data on the numbertheory exam i took on tuesday:
number of students taking exam: 10
mean (average): 23.8 (out of 40)
standard deviation: 6.596211034
median: 26.25
kurtosis: 1.333906232
geometric mean: 22.82177487
harmonic mean: 21.82918952
maximum score: 40 (by the lecturer), 32.5 (by one student)
first decile (i.e., 10% of scores are below): 15
second decile: 15.5
third decile: 16.5
fourth decile: 17.5
fifth decile: 24.5
sixth decile: 28
seventh decile: 28
eighth decile: 29.5
ninth decile: 31

i dont know what i got. i guess ill see tomorrow at 10ish.

Um. 10 students. Professor decides to publish deciles? Seriously? That’s an improper, terrible use of statistics if I ever saw one. Wow.

(more…)

Note: I wrote this years ago, to be precise on 7 January 2007 — so some of the links might be out of wack. At that point I still lived in LA and was a historian. But I thought since I haven’t been posting all of my current fabulousness, I would at least give you some of past fabulousness.

NAVIER-STOKES EQUATIONS

This quest was spurned on by a friend who was interested in learning more about the Navier-Stokes (N-S) equations. (They’re a system of equations, which is why they are referred to in the plural. All the equations together, describing the system in 3-linear dimensions and 1-time dimension, define how fluid flows.)

I learned about them in 18.354, a class devoted to the study of fluid dynamics at MIT. What are the N-S equations, you ask? As stated, the equations describe the way fluid flows – but fluid means more than just things like water, but almost anything from honey (very viscous) to a gas (well, a gas moving at speeds much less than the speed of sound). The only limitation is that the system has to abide by something called the “continuum hypothesis”. One website nicely puts it like this: “The basis for much of classical mechanics is that the media under consideration is a continuum. Crudely speaking, matter is taken to occupy every point of the space of interest, regardless of how closely we examine the material… it is well known that the standard macroscopic representation yields highly accurate predictions of the behavior of solids and fluids.”

So the N-S equations can answer some pretty cool questions about everyday life. Why does an airplane fly? The answer lies in how air flows around its wings. How long does it take for a stirred cup of coffee to become still? The answer lies in the effect of the cup on the coffee. (Believe it or not, the velocity of the coffee very very close to the edge of the cup — called a “boundary layer” — is actually zero. The coffee doesn’t move. And this layer, over time, eventually affects the rest of the coffee spinning until the coffee is totally still. Of course, to be totally precise, you need to take into account the effect of the bottom of the cup too.)

It’s hard to explain what makes the equations so neat. First is that even though they look complicated (see Wikipedia), they are actually pretty easy to derive from first principles (read: from scratch). Second is that they apply to so many phenomena — and much experimental work that has been done confirms it. Third is that they are still pretty mysterious. I’ll get back to that soon.

The N-S equations straddle the boundary between the pure and the applied. To be more accurate, perhaps, they do a good job of demolishing the myth that there is a “pure” and an “applied.” They are used as tools in a number of real-world problems. But at the same time, they represent a challenging problem of pure mathematics. This is what I meant by “mysterious.” They aren’t quite as well understood as mathematicians would like. Right now, the Clay Mathematics Institute has offered a $1 million prize to the first person to make some real headway into understanding the N-S equations: it is a millennium problem.

MILLENNIUM PROBLEM

For the laypeople, not us, the Institute describes the problem as such:

“Waves follow our boat as we meander across the lake, and turbulent air currents follow our flight in a modern jet. Mathematicians and physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an understanding of solutions to the Navier-Stokes equations. Although these equations were written down in the 19th Century, our understanding of them remains minimal. The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations.”

That’s a bunch of fluff. The way the problem is posed by the Clay Math Institute for mathematicians, it appears insanely complicated. But I am going to try to explain the problem for a population who wants more than fluff and less than incomprehensible jargon.

The N-S equations are something called “Partial Differential Equations” (often referred to as PDEs). You might remember seeing differential equations in calculus (e.g. dx/dt=2x). Well, a partial differential equation is like a regular differential equation you’ve seen, but with more variables (e.g. du/dx+du/dy=3). When you first learn to solve them, in your intro PDE class, you are given a hodge-podge of tricks to solve a limited set of those equations. But this isn’t bad teaching; it’s the nature of the beast.

Let me explain. Remember in calculus when you were given an integral, and you had to figure out how to simplify the problem in order to integrate? You couldn’t integrate everything put in front of you… even if you wanted to, you wouldn’t have been able to integrate (x*sin(x)*5^x)^x by hand, because there’s no good way to do that short of putting it in a computer and churning out an answer. So you could only solve by hand certain, easy integrals. The same thing goes for PDEs… they are really tough to solve by hand… sometimes they can be solved by computer… and sometimes there isn’t even a solution! (How could there not be a solution? Sometimes, no solution exists for a problem. For example, the equations “x+3y=7” and “3x+9y=4” can’t give you a solution for x and y. No x and y exist that satisfy both equations.)

So in an intro PDE class, you learn to solve only certain “classes” of PDEs that have solutions. Sometimes you’re lucky and you can do them by hand. Othertimes you can’t and you have to put them in a computer. But, as I said, sometimes there isn’t even a solution.

What the Millennium problem asks a mathematician to prove is “merely” that the N-S equations have a solution (this is called proving the “existence of a solution”). [1] What does this mean?

So say you’re given a fluid system (imagine, say, a really really big cube of water, so large that for our purposes, it is so huge that it fills the universe… in other words, all of space is filled with this fluid) and you are given the following pieces of information:

1. the fluid’s viscosity (in this case, the viscosity of the water; remember viscosity represents the internal frictional forces of the fluid), and

2. the initial conditions of the system (the velocity of the water at every point in the universe at a certain time t=0)… [The initial conditions you are given are “smooth”… continue reading to find out what this means.]

So you have this giant cube with liquid in it, and you know how the liquid is moving at the beginning. You let the liquid continue to move around, defined by the N-S equations which describes fluid flow. Liquid with a velocity pushing upwards, for example, will displace other liquid which will displace other liquid, etc., and the whole system is churning.

EXISTENCE: The millennium problem says that you have to be able to prove that a solution exists to the N-S equations. You don’t need to find the solution, but you need to prove that it exists. What does “a solution to the N-S equations” mean? What it means is that by solving the N-S equations, you can (1) give the pressure of the fluid at any point in the universe, and any time in the future, and also (2) give the velocity of the fluid at any point in the universe, and any time in the future. These two things (the pressure and the velocity) define the system; if you can find both of these, then you have the solution to the N-S equations.

But recall that the problem is abstract. So to solve the problem, you can’t merely say that for a single particular system, you can show a solution exists. It would be pretty easy to show, for example, that a universe filled with water which is at rest at time t=0 (the initial condition of the fluid is still) will never change. So you can say that a solution to this system exists. But you haven’t solved the millennium problem. What makes this problem hard is that you have to say for all systems, solutions exists; in other words, you want to say that for a universe filled with a fluid with ANY viscosity, given ANY initial conditions, a solution exists. That’s what makes the problem hard.

SMOOTHNESS: There is one thing I left out, but now I can add it in. The Millennium problem doesn’t just ask that you show the existence of a solution, but also that the solution is smooth. In math, “smooth” has a particular definition, but what you need to know is that in this problem, the desire for a “smooth” solution comes out of a physical concern. The system, at every point in time, must have a finite energy. (For those who care, mathematically, this is calculated by taking the integral of the square of the speed of all the points of the system, and showing that it is less than infinity.) [2]

COUNTEREXAMPLE: Of course, one easy way to solve the problem is to prove the opposite. I think mathematicians generally are fairly confident that there is a proof that can show the existence and smoothness of a solution to the N-S equations. But if you can come up with just a single system with smooth initial conditions, a particular viscosity, and smooth external forces acting on it (like gravity), and prove that that system DOESN’T have a solution, then you’ve also solved the problem. Because you’ve shown that no matter how hard mathematicians try, they can’t find a proof to the problem, because you’ve find a counterexample.

[1] To make the problem easier, the Millennium problem people even said that you don’t need to consider ANY external forces on the system (so, for example, in the universe, you don’t need to have any gravity). In the most general version of the N-S equations, these are incorporated.

[2] Recall, however, that the initial conditions have to be “smooth.” This fact should make it easier to show that the solution will be “smooth.”

PS. I’m aware that I probably got some of this wrong. Plus there’s the added difficulty of being 100% truthful mathematically while using words without just writing the math out, which pretty much defeats the purpose of me trying to do this. Feel free to correct.

UPDATE: I finally found a really nice explanation of the problem on this blog which attempts to explain a proposed (but now shown wrong) solution. Read up to but not including the paragraph beginning “So, how does Penny Smith’s analysis approximate this by a set of hyperbolic equations?”

So right now I am sitting in Hynes Convention Center – room 109. In case you aren’t in the know (for shame!), I am at the National Council for Teachers of Mathematics (NCTM) conference in Boston. I just finished Day 1. I spoke to a total of three strangers, one of them who I recognized (and who recognized me) from the Phillip Exeter conference from this past summer. I don’t do well with meeting new people, which is such a shame in such a math-teacher-rich environment. But hey, three isn’t bad.

The sessions I went to today were:

#14: Identifying and Remediating Misconceptions [about CAS/TI-Nspire and developing numerical intuition]
#46: Show me the Sign! [about using sign analysis effectively in 9th, 11th, and 12th grades]
#79: Helping Students Read Math [about how to teach students to read their textbooks]
#142: Discovering Trigonometry [on how Exeter uses problem solving to teach their courses, using trigonometry as the vehicle to talk about that]

This was my first NCTM conference. Let me put this one piece of information about me: I don’t like my time wasted, so I tend to be critical of speakers [1]. I expected to really appreciate one or two of the sessions, and politely sit through the others. I thought I’d be inspired maybe once or twice.

You can see where this is going. I really, really enjoyed all four sessions. The speakers were prepared, and focused – for the most part – on concrete things in the classroom. It wasn’t about giving us the most difficult but interesting mathematical problems to work on. In other words, it wasn’t about mathematics. It was about teaching mathematics. We talked about topics and skills we work with everyday, and the speakers spoke about their approaches. None of them were zealots, saying “you should do it my way because it is the best.” It was “this is what I do, this is why I do it, and maybe you can use bits and pieces of what you hear here in your own classrooms.” I appreciated that.

I don’t know if I will have time to post about each individual session, but I will hopefully post some interesting bits later. (I said that about things I learned at the Exeter conference this past summer, and never did, though. So I can’t promise.) But maybe if (when) I actually apply some of what I’m getting to the classroom, I’ll feel more inspired to write.

(FYI, if you feel like you just absolutely need to know more about one of the sessions I went to, throw that down in the comments. You know I can’t deny you.)

[1] Yes, yes, I know our kids feel the same way, and we should always keep this in mind when we enter a classroom.

This year our Math Club members are really intent on training for the AMC. They want someone in our school to break a score of 100 to move onwards to take the AIME.

Here’s the deal.

I want to help the leaders of math club find a way to do this. I don’t know how. We only meet for 25 minutes a week.

Is anyone out there a leader of a math club, that “trains” students for these types of contests? How do you do it? Literally, I’m asking for how you structure a meeting, and what kids are doing, and what you are doing during that meeting.

Also, if you as a math club adviser have any websites or books that you find invaluable, that would also be of great help.

I assume one of the important websites is Art of Problem Solving. After a ton of digging, you find that on that site is a list of AMC problems of years past, and solutions. What else ya got?

PS. One of my favorite math competitions from when I was in high school was the USAMTS. It’s a mail in math contest with 4 rounds, and amazingly wonderfully frustratingly challenging problems. So if you don’t know about it, and you have a super talented math star in your school, I’d check it out and (if you like it) share.

So I got that really wonderful email in my inbox last week, so of course I emailed this student back. I was insanely curious why s/he was taking Calculus in college. Most of the kids in my classes aren’t really interested in pursuing a science/engineering/math degree. So I asked why.

Hey Mr. Shah,
I’m glad my email could contribute to a better day I feel like that’s a very nice accomplishment.  As for the why and how – I decided I had such a good time learning Calculus last year that I should continue taking it in college.  I’m taking [course#] which is a basic Calc course […].  I’m actually really glad that I took Calc last year because sometimes the way my professor explains things makes concepts harder than they really are.  We learned about continuity today and I was happy to see I remembered basically all of it.  I think what we did last year was really good and I think you did a good job explaining things and making sure we really knew what we were doing.  The addition of the youtube videos and things like using our fingers as slope meters helps make things more visual too.  The sad thing about being in a college level class is it’s bigger and the professor goes much faster so you don’t always have those “a-ha” moments (or the time to appreciate them) that we had last year.  I also told [other student from the class] about your email and he said he really misses our old class too and that he’s taking Calculus this year as well and it hardly compares.

Um, at least TWO of the students in that section of 7 are taking calculus in college? Really? SERIOUSLY? ZOMG! I kind of can’t stop smiling. Can I think of any better thank you than that?

No, no I really can’t.

Commence swelled chest.

(That feeling is actually fighting with my feelings of inadequacy and failure which I’m feeling now at the start of this year. It’s a strange place to be in. Like “I must have been good last year, so why am I doing not so hot this year?”

In my inbox this morning:

Hey Mr. Shah!!
I just wanted to tell you that I’m taking Calculus this year and I have to say, it is just not as fun as it was last year.  My teacher doesn’t have fun pictures and she writes on a CHALK BOARD *gasp*.  I miss our class!  Hope you’re having an awesome year so far.

She’s not the only one who misses our class. This just made my week. Seriously.