Month: May 2008

Student versus Teenager

I was never them. I can’t relate.

I was looking online for something related to my school, and I came across a LiveJournal by a student (from my school way before I came). He writes sporadically about his senior year (sometimes writing from class)!

Some things he says:

  • I just did absolutely awful on my bio test and I am really disappointed in myself. Weinsieder just farted and I’m in Stats… this is great. Another wonderful beginning to a day in the best school on earth….
  • Last night was pretty fun now that I think about it. Smokin a fatty and then going to Wo Hop then the Knicks game, sounds like a decent night. The problem was… marijuana is not the drug for the Knicks game. I was about to pass out right in the seat.
  • F*** YOU… I can’t believe you. I can’t believe how you can ruin me in 2 minutes of conversation. I don’t know how you can live with yourself, with just killing me repeatedly… I can’t even describe how I feel and how much I want to punch something but I’m not going to, I don’t want any more scars on my hand and I don’t want any more scars period.
  • Weird a** night. Went to the party… threw up..drank a lot… broke my knuckle…went to Vegas on Smith St… met some girls… went to their crib…
  • I was just thinking about going to Europe and how it is going to be amazing. I hope I survive and don’t go to jail and don’t get poisoned. I’m going to be rebellious and get an ear-ring before I go. My parents can’t rip my ear off if I’m in another continent. I have also been thinking about prom and how I really hope it is somewhat okay.
  • Well hmm I dont know what to say. Today I did a lot of loitering, not knowing what to do with myself…. trying to pretend to sleep. I realized that I have to start playing basketball again because 2 years ago I was pretty good and now I can be better if I get my act together. School is wwwiiiiiiiiiiinddddinnnng down and I am getting more and more aburrido.
I’m a young teacher; it wasn’t so long since I was in high school. But clearly we only see a small fraction of our students’ lives. And reading this journal reminded me exactly of that fact. Even though it often feels like they are our jobs, and even though we see them almost every week day, and even though we’re with them for a year, they don’t see us proportionally. We’re just one small piece of something much larger. We’re just one part of a constellation of teachers, while there are other constellation — many constellations. Of friends, of family, of peers, of lots of things.  I started thinking more about this recently, but after reading these journal entries, I’m expanding how I have to think about them.
Back to the kid above. I started out by saying “I was never them” — which is true. I can’t relate to many of things this kid talked about. My life was incredibly different, and I’ll freely admit that. But what comes through in these posts is more than just the drinking, partying, and apathy.
And so I lied. Because I can relate. The emotion drips through in everything he writes — the ennui, or anger, or angst. The boredom. I remember the boredom.  He cares, when he’s trying to project he doesn’t. He thinks, he feels, he is a creature whose life is a series of contingencies, he is trying to figure out who the heck he is and what he’s all about.
And let’s be realistic: we’re only a (very) small part of that.
We matter, but only a small bit. We teachers see one small slice (50 minutes of class) of something really complex (a whole life). And I don’t think I ever really knew that until this very moment.
Will this affect how I teach? I don’t know yet. Because I’m not sure I teach to mold that young person. Dan Meyer asked this exact question a few weeks ago.
In what two ways will your male teenage students spend their free time and disposable cash this weekend? How much does it matter if you don’t know?
Right now all I’m prepared to say is that: the question has a totally different meaning to me now than it did a few hours ago.
With that said, I entreat edubloggers to go out there and find one or two livejournals written by teenagers similar to those that go to your school.
Note: I kept a livejournal in college; I might do a future post on that. Might.

Algebraic Manipulation Is Overrated

An intuition question.

Look at the function below. It may surprise you that it is a constant! For any value of x, the function g will have the same value. I’m wondering, now that you know this, if you can get a sense of why it would be a constant, without (a) using your graphing calculator, or (b) taking the derivative to show that it is 0 [that is what I did, and as a side note, I have to use this on a test or homework next year].

g(x)=\frac{\sin(x)+\sin(x+a)}{\cos(x)-\cos(x+a)}

Can you find some geometric way to see that?

It took me somewhere between a half hour and an hour of playing around to get it. I can post my solution in a couple days, but right now I don’t have the energy to find a program to draw my solution [1]. But let me just tell you: it’s beautiful. You’ll be stunned when you first do it. Yeah, the calculus way tells you it is a constant, but seeing the “why” is still a mystery. The geometric way takes a bit, but whoa nellie, you won’t regret spending the time!

[1] Or maybe I should claim there is no room in the margin! (JK)

Update: I did finally write up my solution. I quickly did something I never have done before: do my work in powerpoint. It worked fine.

Update: Mr. K solved the problem in 3 minutes and found a way to show the geometric solution. Head over to his very excellent blog to see it in all it’s glory.

Update: Besides mine and Mr. Ks, a third and perhaps more elegant solution is up at 11011110.

Of the three, I think I like Mr. K’s visualization best, even though it might not be a proof in the formal sense.

“Professional Development”

Each year, my school provides each teacher with $100 of “professional development” money. I don’t know exactly why they call it that (hence the quotation marks). For things like conferences, online courses, etc., we have a really great fund to tap into. No one I have talked to has ever been denied money from that fund. This $100 is more of a mystery. You have to submit receipts for it, and it needs to be for things relating to school. I could buy school supplies, for example. Professional development? Tenuous.

And, in fact, each year there’s a book fair with tons of books for students and teachers to buy from. It’s a fundraiser for yet another something or the other. I learned that it’s tradition for teachers to never use their money during the year, and during this week in May, pick out $100 worth of books from the fair to count as their “professional development” money. This practice is so institutionalized that you don’t ever have to take out your wallet to get the books; the people running the book fair just write your name down and the total amount you’ve spent on a piece of paper and you’re done.

Streamlined, and sweet. Just the way I like it. [1]

I’m not complaining. How could I complain about this? But I do wonder why this money needs to be couched in terms of “professional development”? (No matter how broadly you look at it, my Martha Stewart books will never be professional development.) My suggestion: why not just call it a “we like you teachers and we want to give you a little pick me up” perk and be done with it?  I like my school. But for some reason, getting a $100 and being it’s told because “we like you teachers and we want to give you a little pick me up” is just so much more satisfying than “professional development.” So I’ll pretend that’s what it’s explicitly earmarked as and go along merrily.

[1] I know you’re wondering… I bought two very smart-looking hardcover Martha Stewart books (“classic” and “new” recipes), Middlesex (Jeffrey Eugenides), and The Secret History (Donna Tartt). I’ve read the Donna Tartt book before. One of the best books I’ve ever read, hands down.

End game

I had the end game in sight. I carefully planned out all my Algebra II classes so that we could learn the very basics of matrices and systems of equations, and have one last quiz on them, before the school’s official “review days” kick in. (No assessments during those days.)

Everything was peachy keen.

Until I learned that, oh, yeah, the school was taking away one of my classes and giving it to an Academic Awards Ceremony. Which is fine, I can deal. But that’s one of those things that get slipped through the cracks in terms of “let’s tell new teachers that the awards presentation is during class!”

The point is, everything had been planned out. I had dotted every i, crossed every t. (The jots and tittles were there, I swear!) Now with that class given up to the awards ceremony, everything gets totally screwed up in terms of teaching. It’s not just missing that one class, but it’s a perfect storm. One consequence is that, get this:

there will be a stretch of 7 days (that’s including 2 weekend days) that I don’t see one of my Algebra II classes, due to something or another.

Let’s go through them in order: there’s the one day that week we don’t meet regularly (rotating schedule), there’s a high school field trip day, there’s the awards ceremony day, there’s saturday and sunday, there’s memorial day, and there is registration day.

There are other issues, in terms of the quiz I was going to be giving them. Normally, this is all no sweat. I roll with the punches, I can jiggle something here and finagle something there. But in end game mode, you have nothing to adjust to make everything work. I’ll pull a Tim Gunn, and it’ll all play out nicely. It’s just annoying that I have to pull a Tim Gunn in the first place.

As an aside, sorry for all the metaphors, or whatever I’m doing (“roll with the punches,” “end game,” “jot and tittle,” “perfect storm,” “slipped through the cracks”). I don’t know what’s wrong with me.

Calculus Projects! Or, How to Combat Senioritis.

The year is coming to a close and I’ve found something to entertain my seniors. They’re taking regular calculus. More than likely, most of them will never take a math class again. If they are going to take math in college, chances are they’re going to be taking calculus over again (I don’t teach the AP calculus classes at my school).

My school treats seniors with the deference that seniors think they deserve. They don’t have to take final exams, they don’t go to classes after May 22nd (don’t ask), and they miss a lot of May to AP exams. All in all, because of these restrictions, May is pretty hard to plan, if you teach a senior class.

I gave my last quiz recently, and I’m having students use their class time to work on a calculus project.

I only have 7 students in this class, so I decided to do something pretty radical. I pretty much gave them free reign on their project. I told them they could do anything they wanted — just as long as they’re passionate about it. They have to do something they’re going to enjoy doing. They could also choose the point value of the project (a large quiz grade or test grade).

At this point, the only way I’m going to get them to do anything is by tapping into things they like.

So I had them brainstorm, we met individually so I could guide them, and they’re off to the races, with some great projects:

  1. One student is doing a study of Newton’s method (we didn’t cover it in class) to find the zeros of a polynomial. She’s going to compare whether Newton’s method to finding zeros is “better” than a more simplistic method of finding zeros. That method, in case you were wondering, has you find an interval where you know there is a zero (e.g. for example, say you know there’s a zero on [-1,1] because the function is negative when x=-1 and positive when x=1). Then you divide the interval in half (into [-1,0] and [0,1]) and you find which of those two intervals has the zero. Then you divide that interval in half, and find which of those two intervals has the zero. On and on and on…
  2. Another student is doing a study of rainbows, which involves calculus. (Awesome resources here and here.)
  3. Another student really liked learning the intuitive version of the chain rule that I taught (post one and two), and wanted to make a lesson for my students next year on that! So she’s making a video tutorial and worksheet to accompany it.
  4. In the same spirit of teaching, one of my students wanted to do something similar by making a video tutorial on the formal definition of the derivative.
  5. One student is taking AP Physics B, but throughout the course, has noted connections between what he’s learned in his non-calculus-based physics class and what we’re doing in calculus class. One connection he made was between Pressure, Volume, and Work. He (rightfully) noted that W=\int P dv. So he’s going to be making a presentation on this relationship by doing a bit of research and bringing application to the class.
  6. Another one wanted to learn something “new” so I suggested he do some research on a hanging string. More notably, if you hold up a string (like a necklace), it will hang down due to gravity. Surprisingly (or not?) the shape is not a parabola. It turns out that it’s this funky shape called a catenary. He’s investigating why that’s the case, and how to derive the formula.
  7. Last but not least, one of my students had difficulty with the sections on surface area and volume, because she couldn’t visualize the regions/spaces being formed. So she’s making two mechanical thingamajiggers out of wire. You bend the wire to be whatever function you’re going to be rotating, and then there’s a handle that rotates the wire. I am so excited about this one — I hope it works out so I can use the model next year in class!

6MMM^3

The 6th Monday Math Madness is online now. This week is actually not so hard, even though there are two different problems… I was able to get both of the answers in about 30 minutes (assuming I didn’t make any huge errors). I especially like the first question, because it can be easily transposed into a slightly more difficult and fun problem…

The first question reads:

  • Start with 500 gallons of mayonaise.
    1) Mix in 10 gallons of ketchup. Stir until completely mixed.
    2) Remove 10 gallons of the mixture.
    3) Repeat steps 1 and 2 until the mixture is approximately 50% mayonaise and 50% ketchup.How many iterations will it take to do this?

Here’s the slightly modified problem:

  • Start with 500 gallons of mayonaise.
    1) Mix in 3 gallons of mayonaise and 7 gallons of ketchup. Stir until completely mixed.
    2) Remove 10 gallons of the mixture.
    3) Repeat steps 1 and 2 until the mixture is approximately 40% mayonaise and 60% ketchup.How many iterations will it take to do this? 

That slight change makes it more difficult! But fun fun fun!

Two cows are in a field…

In math club this past week, we didn’t have anything to work on explicitly. So we just made up a problem, based on a problem we encountered in the previous week.

Without further ado, here it is. You have a circular field, enclosed by a fence. Two cows Antonio and Barry graze in the field. They are each tethered to some place on the circle, tied with ropes of lengths r_A and r_B respectively.

The problem is: come up with a formula for the area of the region that both cows can graze together.

I love that we came up with the problem, and that we’re exploring it ourselves. It’s great that it’s so simply stated, and that it has a pretty tough solution. I love that it’s a generalization of something we did earlier. And I love that even this problem can be generalized further (e.g. we have n cows).

What we did in 15 minutes:

We know we’re going to have a piecewise function of three variables. To start the problem, we make the circle a unit circle, we place Antonio at the point (1,0) and we place Barry at (\cos \theta, \sin \theta).

By the end of our math club meeting, we had one part of the piecewise function f(r_A, r_B, \theta). We found where there would be no overlapping grazing area, where the function would be zero.

I have some sketches of the problem and the bit of solution we got together. I’ll put them below in a bit.