Monthly Archives: March 2008

So I didn’t *win* Monday Math Madness. For those of you not in the know, the blog WildAboutMath is having a rotating math problem contest with another website, and I put my hat in the ring. And I got the answer right. Sadly, the random number generator did not love me (click link for a few different solutions).

This week’s problem was:

A popular blog has just three categories: brilliant, insightful, and clever. Every blog post belongs to exactly one of the three categories and the category for each post is selected at random. What is the probability of reading at least one post from each category if a reader reads exactly five posts?

After solving it, I tried thinking of natural extensions. Clearly, one is: “if you have p posts and n tags: what is the probability that a reader reads at least one post from each category if a reader reads exactly p posts?” Or one could think about the more difficult question, where the frequency of each tag is different (not equal chance of stumbling across each post).

My submitted solution after the fold. You can see how hard it is to explain math clearly in an email.

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It’s 10:45pm and I just came across this compelling article called “The Mathematician’s Lament” that I’ve now read half of. I’m going to finish it off when I have more time, and post my reaction to it. From what I’ve gleaned thus far, this article is incredible provocative. I can’t say I wholly agree with everything the author says, but he also does hit a few points home.

Some juicy quotations to get you hooked:

Sadly, our present system of mathematics education is precisely this kind of nightmare. In fact, if I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done— I simply wouldn’t have the imagination to come up with the crushing ideas that constitute contemporary mathematics education.

By concentrating on what, and leaving out why, mathematics is reduced to an empty shell. The art is not in the “truth” but in the explanation, the argument. It is the argument itself which gives the truth its context, and determines what is really being said and meant. Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity— to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs— you deny them mathematics itself. So no, I’m not complaining about the presence of facts and formulas in our mathematics classes, I’m complaining about the lack of mathematics in our mathematics classes.

Some good finds I’ve been saving up. Click below for a hodgepodge.

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Trigonometry is one of those topics that if you get the basics, the rest of it will make a heck of a lot of sense. But if you miss it, you’re going to be trying frantically to come up with ad hoc ways to understand each new concept.[1]

I have been teaching the beginning of trig, and I’ve noticed a few things that I have to watch out for next year:

1. Have a really good reason prepared for explaining why we care about angles greater than 360. In general, the kids don’t have a good idea of why we’re doing what we’re doing. (“Why do we want to find csc(421)?”; “Why don’t we just say we have an angle of 1 instead of 361? When would we ever need 361 degrees if a circle has only 360?”)[My explanation didn't hold over well, but it's true. I said that often times we use angles to measure time, like in a clock. So if we have something repeating -- like a spring with a mass in physics, or a ferris wheel going around and around, or a bike wheel spinning -- we will be able to model how far it's gone or how many oscillations its made by using this angle.]
2. In fact, have a good reason to explain why we care about angles greater than 90. I started out teaching triangles and SOH CAH TOA, and they got it. Then I started teaching how to see angles on the coordinate plane, and I lost some of them. They can — I hope — calculate the sine of 210. But they don’t get why the sine of 210 is at all related to the sine of 30. They see the 210 angle, and they say “where’s our triangle”? And I show them the 30 angle, and they understand that we can form a right triangle with it, but they don’t get why we use the triangle with 30 to deal with the 210 angle.[My explanation dealt with looking at triangles made in the first quadrant, like the sine of 45. I showed them that the opposite side of the triangle was the y-coordinate, the adjacent side with the x-coordinate, and that the hypotenuse was the radius. Then I said the problem with the whole "opposite," "adjacent," "hypotenuse," method of things was that it restricted our angles to lie between 0 and 90. So to expand the domain of these trig functions, we put them on the coordinate plane, and defined sine to be y/r, cosine to be x/r, and tangent to be y/x.

   But then they asked: "Why? Who cares about angles greater than 90?" Which then takes us back to #1.]

3. I haven’t given them a big picture, which is part of the problem. Right now they’re learning smaller skills, but they don’t know what the whole point of it is. So what if you can find cot(260)? Why do we care? What does cotangent mean in the real world?

Next year… I might want to motivate trigonometry on the first day, and give a hard application problem that we’ll be able to solve by the end of the unit. Then I’ll give an schematic diagram of what we’ll be doing and try to motivate each step. That will give them the big picture. And hopefully the rest will fall into place once we have the big picture.

[1] It’s like that old adage… “I must’ve been absent that day.” In this case, it’s pretty disastrous.

UPDATE: A student asked me today, “Mr. Shah, will you promise me something?” “Not without knowing what it is.” “Well, will you promise me that you’ll explain why we’re doing all of this at some point?” So I was right on the ball, in terms of students just not knowing what’s going on because they can’t see the forest for the trees. And I went looking in the book to see what applications they have, and they’re awful (kites, ferris wheels, and bicycles). I’ve got to get on this right away!

I got roped into chaperoning a school dance.

Okay, you caught me. I was hoping for the opportunity since I started teaching. Why? I want to act as sociologist, studying the behavior of my students in a setting as much outside of the educational context as I’m going to see them. (Yeah, being in NYC, I see my students on the subway at random times and places, but that’s not this…) Adults aren’t acknowledged, at least not at my school dances. I and all my friends ignored all our teachers at homecoming and prom. I know what it’s like to be on the other side of the teacher’s desk. Now I’m curious what it’s like on the other side of the dance floor…

One teacher said that after she chaperoned a dance at her previous school, she couldn’t look her students in the eyes for the rest of the year.

Wow.

I’ll report back with general observations, if I have any. Stay tuned.

Update: The night wasn’t scandalous, and chaperoning was just a lot of standing around. The few observations I had:

1. Old school Britney (“Toxic”) will get all students to the dance floor.
2. Old school Britney will get the chaperones who are assigned the dance area to dance. (Yes, that was me.)
3. Cutting off a poppy-electronic-song with a female vocalist (like many on the videogame Dance Dance Revolution) will cause howls of execration.
4. 4. Playing Soulja Boy’s “Crank That” is the closest you can get to having a “High School Musical” spontaneous song-and-dance-number moment.

5. You can earn bonus points with students if you find a tube of lip gloss at the end of the dance, and ask them: “Hey, this lip gloss belong to anyone? It be poppin’” [see this video for explanation]

Last week, I proctored the American Mathematics Competition (AMC10/12). Three students in mathclub showed up to take it, along with 4 other students.

When I was a student in high school, I loved these tests. Not only did the 25-question-test set my brain on fire, but the problems were actually do-able. I loved math competitions (e.g. the USAMTS, the New Jersey Math League contest). Not only did I enjoy those types of problems, but I enjoyed the competition aspect of it all. I liked that I was competing not just with myself, and others in my school, but also people nation-wide.

I didn’t excel much in high school. I mean, I got good grades, but I wasn’t an amazing writer, artist, computer programmer. But I did have math — that was mine. It was part and parcel of my identity.

After the test was collected and sealed, students tallied up their scores. [The scoring works as follows: 6 points for each correct answer, 1.5 for each unanswered question, 0 points for each wrong answer.]

Unfortunately for the mathclubbers, you need 100 points to move to the next round of the math competition (the American Invitational Math Exam — AIME), and none broke that barrier. So this year, our students are out of luck.

As a promise to the mathclubbers, I told them I would take the exam at home, under testing conditions (75 minutes, no breaks, no calculator). I got 16 questions right, no questions wrong, and left the rest blank. That gives me a score of 16*6+9*1.5=109.5, which would have qualified me for the AIME. In high school, I always scored around that level.

(And I was nervous that these high schoolers would school me on this test.)

The score is bittersweet. It meant that I still “have it” (“it” being the ability to do AMC problems). It also means that I have not grown smarter since high school.

This post on the site “stuff white people like” is just so dead on that it’s hilarious. Not the whole “white people” part of it (I’m not white), but everything else. One of the best quotations:

At this point, they can feel superior to graduate school and say things like “A PhD is a testament to perseverance, not intelligence.”  They can also impress their friends at a parties [sic] by referencing Jacques Lacan or Slavoj Zizek in a conversation about American Idol. 

 That’s (embarrassingly?) exactly how I feel about a PhD. And what’s even sicker? (No, no, I haven’t invoked Lacan when talking about Idol… or any reality show.) Today we had a middle and upper school “instrumental” (which means that the school’s various musical ensembles performed for all of us). And I was sitting next to one of the dance teachers. And she commented on how much the conductors’ movements were like dances — expressive and distinct for each one. And I, appropos of the quote above, responded with a comment about “embodied knowledge.” 

Working towards (and deciding to stop working towards) a PhD has forever transmogrified me into an intellectual snob.  –He says, as he glances over at the book (Tom Clancey) he’s reading, the DVD case (Gilmore Girls) of what’s in the DVD player, and the giant red plastic lobster that’s on top of his bookshelf.

Recently in calculus, we’ve been working hard on u-substitution to solve integrals. Integrals are not intuitive. They are motivated (area under the curve), they are justified (the anti-derivative), and we try to play around with them until we “get them.” But when it comes down to it, you can’t really capitalize on much intuition when introducing them.

In my class, the most we did with honing intuition was to ask basic questions like these:, , , , , and .

To solve them, students had to “work backwards” from their derivative knowledge. They were just guessing and checking to see what functions they could take the derivative of to get, say, . (In this case, it was ). They figured out some general strategies on their own, and I validated them.

And then we moved on to u-substituion, to solve integrals like and .The hardest part of u-substitution is picking a good u which actually simplifies the integral. (In the first integral, and in the second integral above, .) The question for students is: why? It would be a mistake to assume that any explanation you give them will make sense (initially). For the second integral, saying–

“Well, because I see that square root on the bottom, I immediately think of inverse sin, because we know . Hence, I see that we should pick a u which makes the integral take this form, and hopefully replacing dx with something in terms of du will cancel out the numerator.”

–will never work. You can say it until you’re blue in the face, but until they try out a whole bunch of possible us which get them nowhere, until they’ve seen enough similar questions like this, until they try to articulate for each integral why they chose that particular u, then they won’t gain intuition. You can’t force it. It comes through familiarity.

This is one of those topics where I think new math teachers probably dread introducing (and if they aren’t concerned about it, they are better people than I am). Students invariably will be confused. And what’s worse (from the student perspective) is that there isn’t a procedure or routine to get the answer. It involves guessing (a u). Educated guessing, but guessing none the less.

A few students have been freaking out — they just don’t “get it.” But I promise them that if they keep at it, it will work out. But they need patience, practice, and to reflect on each step of what they’re doing (why am I picking this particular u?). I’ve designed my lesson plans around these three elements: we’re going slowly, we’re doing tons of problems, and students need to be able to articulate some reason why they chose a specific u. I want them to know that this is a process they need to go through to come out better on the other side.

As Math Stories eloquently puts it:

Learning is not easy. If it was, we wouldn’t need schools and teachers. My kids frequently say that something I’ve taught them is easy. That’s once they’ve learned to do it – it’s the reward at the end of the process. While they’re learning it, they whine and complain and get headaches and have to use the bathroom and everything else imaginable. But that knowledge can only be integrated into their heads by experience, by wrestling with the fundamentals, by trying it out and repeating it and seeing how it works together with other things they know. They may eventually learn shortcuts that make things easier for them, leaving the methods they cut their teeth on in the dust. There are many teachers who will jump straight to the easiest methods, because that’s how the students will end up having to apply it in the real world. But jumping ahead, providing easy technological solutions for things they would otherwise struggle with, is just robbing them of an opportunity to really really learn.

Except for that one unnecessary word (“technological”), this quotation is exactly how I feel now. I will probably show it to my kids too.

Related Posts: Calculus Intuition I, Calculus Intuition II

Yesterday I watched “King of Kong,” and a while ago I fell in love with “Spellbound.” Both of those movies are documentaries about strange subcultures of people — where the norms and values of these subjects are so foreign to the viewer that it’s a bit of an anthropological expedition.

While watching “King of Kong” I decided that a movie needs to be made about a mathcamp somewhere.

Some backstory: I went to mathcamp. Twice. And I was a counselor there once.

Those summers (especially those during high school) were transformative. I was one of those kids who were freakishly [1] good at math, astounded by the elegant beauty of it all, and I just got it. And I was always hunting for more. I would pick up second hand math books and study them, did the AMC (then AHSME) competitions, and did the write-in USAMTS competition. I was on my first high school’s math team (my second high school didn’t have one). And still, that wasn’t enough. I wanted to go to a place to study math for 5 weeks.

Seriously.

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Dan Meyer asks the question:

Unless my experience as a classroom manager is several deviations below the mean, other people are struggling with this as I have struggled. New teachers are struggling with this. So why is classroom management the farthest topic from anyone’s blog?

Before reading the comments, to see what his other readers thought, I suspect that others will agree with me: talking about classroom management meaningfully often times requires speaking in specifics about individual students or incidents. For those teachers blogging under their own names, there’s the added thought: “what if…”

What if students come across the site (probability: likely)?

Don’t get me wrong. I think a lot of value can be had by sharing these stories, getting advice from others, and just commiserating about the difficult moments that come up in the day-to-day. But doing so publicly makes it harder, because specifics have to be pitched out the window. (I don’t want a student coming across my blog, knowing a post is about him or her, and feeling uncomfortable.) And for an issue like classroom management, it’s all about specifics. The individual student, a particular incident, a conversation or punishment. Without that, it’s all and all (just) another good teaching tale.

That’s not to say that conversations about techniques on how to keep a classroom running smoothly and effectively aren’t worth having. It means that talking about when a classroom isn’t run smoothly is harder.

That being said, I’m going to hopefully give Dan something to play with… I’ve typed up a list of notes I took before starting teaching this year, given to me by a veteran teacher: Advice to New Teachers on Classroom Management. I normally eschew prescriptive teaching talk (someone telling me this is the way to do things), but these tips are so useful that I ignored my initial gag reflex and I’m a better person for it. (I’ve noticed that I do a lot of these things naturally, which is a good sign for me, I think.)

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