What post-college life is like (nutshell version)

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.

How knowledge becomes tacit, or how my calculus students rail against u-substitution

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:\int 5dx, \int x dx, \int x^3 dx, \int x^\frac{1}{2}dx, \int \frac{1}{x^{50}}dx, and \int \sin(x) dx.

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, x^\frac{1}{2}. (In this case, it was \frac{2}{3}x^\frac{3}{2}+C). 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 \int (2x-9)^5dx and \int \frac{10x}{\sqrt{1-4x^4}}.The hardest part of u-substitution is picking a good u which actually simplifies the integral. (In the first integral, u=2x-9 and in the second integral above, u=2x^2.) 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 \frac{d}{dx} \sin^{-1}(x)=\frac{1}{\sqrt{1-x^2}}. 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

Recollections of theorems past

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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Why? I’ll tell you why I have to remain silent…

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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My First Post

I’ve been blogging about my first year of teaching at my school in Brooklyn. But until now, the blog has been under wraps, private, an archive of the good, the bad, and the ugly. As I’ve been getting more and more into reading math teacher blogs, I’ve been getting more and more antsy to contribute to the conversations they’re having.

And so with no further ado, I here start my own public blog, and here is my first post.

“Wait! Wait, Mr. Shah! This blog has a ton of posts before this one.”

(Ah, ever the astute reader.) Yes, I’ve imported about half of my private blog here. The other half will have to remain private.

teachable moment, ahoy!

Today I had one of those “teachable moments” that teachers talk about, when the stars align, the opportunity presents itself, and you deviate from your lesson and really capitalize on what’s going on around you.

On Friday there was an incident at my school; a threat was made to one of the students, and there was an implied threat to the community at large. The school was put on lockdown (which means that students weren’t allowed to leave for lunch). Without going into details of the specifics, it caused a bit of panic among members of the community who didn’t know what was going on or why (students, teachers, and parents).

Today, the teachers had a meeting before school, the students and the teachers had a community meeting at the start of school, and then in homeroom advisers were instructed to talk about the incident with their advisees.

My homeroom didn’t really want to talk, much, which is fine. However, when my first period calculus class met, they were in a tizzy to talk about it. They said they didn’t get to finish what they were talking about in homeroom, or their homeroom teacher didn’t really let them talk about it, and they wanted to have that conversation.

I put down my book, and we had today’s “teachable moment.” I let them talk, I would sometimes interject my opinion or get them to think about what they were saying critically, I didn’t defend or attack anyone’s thoughts, I just let the dialogue unfold. Students talked to each other, got things off their chests, and really thought about the situation. Finally, when they were done, I suggested that they write down all their thoughts and feelings about this incident and how it was handled by the administration now before they forget. Those views will be useful when the faculty meet to talk about the administration’s handling of the threat.

Yeah, we didn’t get through some basic integration stuff. But guess what? This. Was. More. Important.