Month: September 2010

Writing College Recommendations

I actually spend a long time crafting college recommendations. I start each one from scratch and I don’t cut and paste. Generally, the way I start the process is I think back about the student, and come up with a few adjectives that sort of define my mental picture of that student. I jot down any specific memories that might be special, interesting, or funny. (With my horrible memory, though, these are few and far between.) I then look through my gradebook and see how the student performed, and any improving trends or points where the student shined. Finally, I’ll look through the narrative comments that I write twice a year on the student.

There is one more thing, and it’s one of the most important things I do, when I write recommendations. I ask students to write a recommendation for themselves. I give them some guidelines. I’m paraphrasing, but it goes like this: “Don’t be modest. Do be specific. Do talk about your feelings about mathematics (and if they’ve evolved). Do talk about if you like math (and if you do, what is is about math that you like). Do tell me if you have a distinct or favorite memory from class. Do tell me if you worked outside of class with other students? Do tell me something you’re proud of.” Basically, they know themselves better than I know them. And they know details I may never know. Like how — for example — a friend might have called for math help at 10pm at night, and this kid spent an hour talking math on the phone.

(Some teachers make a form with narrative questions for students to fill out. I might switch to doing that, although there is something really nice about giving them the onus to determine what they feel is important. Not what I feel is important)

Here’s the three big things that I think are important in a recommendation:

  • The recommendation should be specific, and full of details.
  • The recommendation should  paint a picture of the student — not be generic tripe.
  • The recommendation should be honest– not hyperbolic.

It’s hard, though. Especially for the students who didn’t stand out in class academically. The middle of the pack students. But for those, I don’t lie or slightly puff up the truth. I focus on the character and fortitude of the student when the going got tough.

Some sample paragraphs from recs I’ve written — with names/details changed:

Ex. 1) Historically, most of my students in this class tend to work methodically and systematically, and think inside the box. Stu thinks in a way that I can’t fully understand: incredibly visually and incredibly intuitively. He can “see” things in a way different from the others—and even myself. It is incredibly helpful for Stu in problem solving—he often uses this intuition to make the crucial connection that leads to the solution. In the beginning of the year, Stu found it difficult to express his ideas in words, to translate what was going on in his mind to a clearly written solution. That is a struggle that most of my students go through, never having written a two page solution to a single problem before. However, as he has been asked to write more, his skills in this area have improved greatly. He now knows how to better express his ideas so that they are firmly grounded in the logic of mathematics and so they are understandable to a reader.

Ex. 2) Stu is also one of the most thoughtful, respectful, kind-hearted students I have encountered. When doing group work, he focuses the group on the task at hand and makes sure that everyone is on the same page. When entering the classroom, he always gives an upbeat greeting. He brings lightness and humor to our room. And when leaving, he says “thank you.” Honestly, what student consistently thanks their teacher after class?

Ex. 3) Stu works incredibly well with others in our small class. There is an emphasis on collaboration and working through ideas together. It is a class where students play off each other, help guide each other along, instead of me explaining everything to them. Although Stu almost always has his “a ha” moment before the others, he spends time making sure the others have that moment too. He is a natural leader of the class, propelling us forward without leaving anyone behind. Stu also grapples with new ideas, making sure he mulls them over to get a sense of how they fit into the larger scheme of things. When he asks questions, they usually either anticipate topics we’re about to cover, or drawing connections to topics we’ve previously covered. These questions raise the conceptual component of our class considerably, because they get all of us to think deeply about ideas.

Ex. 4) When I wrote in his first quarter comment that I should photocopy his notebook for my own records, I didn’t just mean that hyperbolically. Taking notes and learning how to do homework are skills that students tend to struggle with in 10th and 11th grade math classes. They aren’t skills that are explicitly taught. I decided that this year, I would spend some time teaching my Algebra II class these skills.

With that in mind, I flipped through my mental rolodex of previous Algebra II students; Stu’s name was the name that popped out at me. His organized and meticulous notes, his step-by-step homework, and his mature approach to my class were exemplars that I knew could help other students. I invited Stu to come to my class this year as a guest lecturer, to talk about the strategies he employed to be successful in Algebra II. He even met with our school’s learning specialist to help him organize his presentation – which was a success. Students really responded to his ideas, and I myself learned a thing or two about how some successful students work. (And, in fact, I did scan in a few pages of his notebook to show students what they should be striving for.)

Ex. 5) I was thrilled when Stu asked me to write a letter of recommendation for her college applications. I had the pleasure of teaching her in Algebra II last year at Packer Collegiate Institute, an independent high school in Brooklyn, New York. In this class, Stu impressed me, not just by her mathematical maturity, but also by the maturity with which she approached school. She is a serious student, and a serious intellect, who – I am paraphrasing her own words slightly – enjoys learning for the sake of learning, because it changes the way she sees the world. It is this attitude, where school is more than just an amalgam of exams but instead countless opportunities to learn new things, draw new connections, and be exposed to as much as possible as often as possible, which makes me certain that Stu will succeed in any institution she attends.

Ex. 6) In short, Stu had changed. Far from the more reserved Stu I had met at the start of the year, the new Stu took more risks in class, and found effective ways for him to learn the material. I don’t see this as simply the transformation of an Algebra II student. I see this as the intellectual maturation of Stu, and the transformation of a young man who has learned that his determination and his hard work can help him realize his awesome potential. He certainly did in Algebra II.

CalcDave you’re the best

CalcDave posted some awesome questions to ask calculus students — to get them think of the very large and the very small… and I made a worksheet out of it. For posterity, I wanted to save some of the responses.

The least probable (but still possible) event that you can think of

  • Going skydiving with the president
  • That I will drop out of high school the day before my graduation
  • When I call ‘stop!’ my watch reads 12:00pm and 0.0000000001 seconds
  • Mr. Shah becoming a rock star in a band called “Pain in my asymptote”
  • A monkey typing a Shakespeare play on a typewriter
  • The Boston Red Sox winning the American League East
  • All the people in the world dying at one time
  • Winning the lottery
  • Pauly D not having a date and Snookie beating Jwoww in a fight
  • The Situation never having STDs
  • A 7.8 (richter scale) earthquake in NYC

Largest number


Draw the ugliest and prettiest functions

I asked the last question about pretty/ugly functions, because I assumed that most kids would draw continuous everywhere and differentiable everywhere functions. And for the ugly ones, those would be violated. We’d have asymptotes, holes, and non-differentiable points. My assumption was realized. So we’re going to have a talk about the aesthetics of math, and coming up with mathematical descriptions for “beautiful functions.” I want them to think about continuity and differentiability, without knowing the terms explicitly.

Now it’s going to be great. Whenever we start talking about infinity or infinitely small, we’ll have some juicy stuff to dig into — stuff they’ve mulled over. Even today, I was talking about watching a video of someone diving and pausing it. And then going to the next frame — and infinitesimally small amount of time afterwards.

We also zoomed in on a point on a graph a huge number of times. An almost infinite amount of time.

And the thing on the screen turned to look closer and closer and closer and closer to a straight line. But it never became a perfect line. Every point on the screen, as you zoom in, gets infinitely close to lying on a straight line. But it won’t ever be a straight line.

So great conversations. We’ll expand them as we continue. Especially how every (continuous) curve is an infinite number of infinitely small line segments joined together.

Random Updates From the Front Line

0. A few students I taught in Algebra II last year came up to me this year and told me that they’re using my binder system in their math classes this year to stay organized. One wants to show me his binder to see how organized he is now. Organization was his Achilles Heel last year. Just hearing that was vindication enough. Because I’d guess about 1/2, eh maybe 2/3, of my Algebra II students last year came to me unorganized. They just had never been taught to correct and organize their work. I had hoped to introduce them to a skill that they might find helpful in the future. So hearing that caused my chest to puff up, my heart to swell. It may indeed have been a heart attack. But what a glorious way to go.

1. I suspected that I would experience cognitive dissonance, because I’m doing Standards Based Grading in Calculus, but not in Algebra II. That — indeed — has panned out. The one SBG-style change I’ve made in Algebra II is that I’m entering things in the gradebook by topic now for Algebra II (e.g. “Compound Inequalities” and “Absolute Value Equations”). At the very least, I can see very quickly in what areas each kid is successful, and similarly, where each kid needs to work.

But here’s the rub. I’m so transformed in the way I’m thinking about student learning, and what assessments mean, and what grades mean, that I am already frustrated that I don’t have a good response to the Algebra II student who comes to me upset about their test grade. Last year, I would have calmed the kid down, and had a talk about the causes of what might have gone wrong. Studying poorly? Not following things in class? Not enough sleep? Whatever. And then we would have made a plan for the next assessment. We were always looking together towards the future.

And that conversation is important. Because part of being a teacher, and highlighted by SBG, is that students need to be hyperaware of their own learning, and proactive in their approach to it.

However, when I’m having these conversations with students this year, I want to also say: “Oh, you’ve figured things out? Prove it. So what if you got a C on the assessment. Show me you know it, and I’ll have your grade reflect that.”

3.  One of my concerns before embarking on the SBG express (woot! woot!) is that I wouldn’t know how to grade on the SBG system. What a 4 is versus what a 3 is versus what a 1 is. But you know what? That’s actually not hard at all, with my rubric. Whenever I get stuck, I ask myself: “From what they’ve written, what level of understanding do they have?” That question settles it. I have a copy of the rubric posted at my desk, and there’s a poster of it hanging in my classroom. To reinforce the importance of the rubric, I photocopied it on the first assessment, so students could refer to it.

4. I’ve raised the bar in calculus. It used to be that on an assessment I would give a bunch of problems, to suss out a level of understanding. Like, I might ask students to find the vertical and horizontal asymptotes, the x- and y-intercepts, any holes, and the domain to the following three rational functions: f(x)=\frac{1}{(x-2)(x-3)}, f(x)=\frac{(x-2)(x+3)}{(x-2)(x+5)} and f(x)=\frac{(x-2)(x-3)^2(x-5)}{(x-2)(x+1)^2(x-1)}. Now I only give one question on the skill, but it tests all I need them to know. That would be the last and hardest rational function, in this case. And to get a 4, I demand perfection.

5. In calculus, I don’t feel any of that “guilt” when a student does poorly on an assessment [1]. You know that moment when grading a non-SBG test, and there’s a question worth 5 points, and you see that the student factored correctly but couldn’t do anything else… and you know if you were being honest with yourself that you shouldn’t give any points, but you think “oh, maybe I’ll give one point just because the student showed me he/she could factor.” You know the thought. Right? RIGHT?!? Well, I have those moments. (FYI: I always go with that is right, rather than what is in my heart.)

With SBG, all that moral teetering goes away. My heart and my mind are finally in sync. I give a student what they deserve and then say “You’re there, you need to be here. Let’s help you find a way to climb that mountain.”

6. Students have started reassessing. So far, not a lot, but I anticipate next week to have — oh, I don’t know — maybe 10 or so? (Reminder: I teach less than 30 kids in calculus.)

7. I tried groupwork & presentations  in Algebra II. It failed. Probably worth another post, if I feel like writing about it. Nothing really exciting. I tried to teach absolute value inequalities by having students muck around and come up with patterns and then generalize and come up with general procedures. I was impressed with the students’ abilities to get to where they did. But it was my organization of the groupwork, and my facilitation, and my rushing because of time pressure, that made it less efficacious than it should have been.

8. I got two really nice emails since school has started: one from a college counselor who said I wrote really great college recommendations, and one from the head of the Upper School who said I had really great course expectations for my students. It’s amazing how a few kind and genuine words can go a long way.

9. Three photography students are putting up an art instillation throughout the school. It’s an amazing idea. They are having faculty each write a short paragraph about why we teach. Then they take photographs of us and post them around the school with our blurb. I had my fashion shoot today, and I’m excited to see what results. If you are so curious, you can read my paragraph here:

Initially, I became a teacher because I loved mathematics so deeply that I wanted to share its beauty with others.. Since I started teaching, though, I’ve come to love something else. I relish the delicious challenge of getting someone who doesn’t know something to actually know something. The thought of changing someone’s view of the world… jut a little bit… therein lies the new thrill.

10. I have to be cautious about SBG. I love systems. I have to make sure that SBG doesn’t just become a system of test, reassess, test, reassess.  It’s not something that students should feel is mechanical. But a process that students go through to learn about themselves, as people and as learners.  And armed with that, they can be proactive and achieve anything. (Sappy, I know.) So here’s a reminder for myself: talk explicitly every couple weeks or so about what we’re doing in this class. It’s not a system. It’s not a system. It’s not a system. It’s a philosophy.

[1] That’s not to say that I don’t think about my teaching, and my teaching’s role (both positive and negative)  in my students’s learning. That’s a given. I’m talking about something else.

Riding a Flying Magical Unicorn

In multivariable calculus today, we were talking about the scalar triple product. It blows my mind that if you have three vectors:

\vec{a},\vec{b},\vec{c}, then you can show that the volume of the parallelepiped defined by them will be:

\vec{a}\bullet(\vec{b}\times\vec{c}). And that if you expand this out, you get:

\mathbf{a}\cdot(\mathbf{b}\times \mathbf{c}) = \det \begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \\ \end{bmatrix}.

I mean, it makes sense that it’s symmetric in terms of \vec{a},\vec{b},\text{ and }\vec{c} — so that no one vector is privileged over another.  But it still seems magical. I mean, even when I know that the determinant of a 2 \times 2 matrix gives you the area of a parallelogram, it still seems so special that the determinant of a 3 \times 3 matrix can give you the volume of a parallelepiped.

So we worked on proving that fact, with what we know about vectors.

There are two hard parts to this proof. One is understanding that the volume of the parallelepiped is the same as the area of the base times the height. (And teasing out what the “height” actually meant.) At this point, I whipped out Cavalieri’s Principle.

It was in this discussion that one of my kids said the most awesome thing. When talking about Cavalieri’s Principle, he said: it’s like if you had 10 reams of paper all stacked up to make a cube. That has a certain volume. Then you push the stack so it leans — maybe so it’s a little curved. What’s the volume of the new stack of paper? The same.

So we understood what the “height” of the parallelepiped actually referred to.

The very last question: how do we find that? And in fact, my kids actually figured that part out without any help. They noticed it involved the projection of one vector onto another.

The rest? Just algebra.

It was lovely. And a different student exclaimed: “You should go around to the various precalculus classes, sharing that what you learn there actually will show up later in life!” (He was referring to vectors, determinants, and even parametric equations.)

Now to the title of this post. I was then transitioning to talking about straight lines in 3D. And I wanted to highlight that you just need a point and a vector to uniquely determine a line in 3D. Somehow, I got it in my head that I should explain it in metaphor. So…

I said that — suppose you are riding a horse. I mean unicorn. That can fly. But only in the direction of it’s horn. And you are told to go to a particular planet and wait there. At the starting gun, the magic, flying unicorn takes off, flying in the direction of its horn.

I am 99% sure this didn’t help kids “get” it. It was pretty obvious to them that a point and a direction (vector) uniquely determine a line. But I really enjoyed talking about the flying unicorn. I liked it enough that I think the flying unicorn may be our mascot for the year.

Plus, I like sparkles. And unicorns have sparkles coming out the…

Intermediate Value Theorem

I wish we could do “baby posts” that were formatted different. Little asides that don’t warrant full posts.  Like this one.

Anyway, today, the AP Calculus BC teacher (and all around awesome person) asked me if I had any good ways to introduce the intermediate value theorem.

That’s the most boring theorem ever. Saying that if you have a continuous function f(x) on [a,b], and u is between f(a) and f(b), then there exists a c in [a,b] such that f(c)=u.

In other words, if you have a continuous curve that goes from point (x_1,y_1) to (x_2,y_2), then at some point along the curve’s journey from the first point to the second point, it’s going to pass through every y value between y_1 and y_2.

If you still don’t see it, just draw two points on a coordinate plane and try to connect them with a continuous function. You’ll see it then.

Anyway, it’s boring. So she was right to ask for ideas. I searched and found none.

So I suggested a warm-up for the class — before they know anything about this theorem. I asked her to throw this up on the board:

INDIVIDUAL CHALLENGE: I am so wise. I have drawn a function f(x) on [1,5], with 10 between f(1) and f(5), such that then there does not exist a c on [1,5] such that f(c)=10. Are you as clever?

And then I wrapped up some Jolly Ranchers for her to give to the first student who could do it.

She said it went really well. And it took a few minutes (read that: minutes) before the first student got it. Perfect warm-up.

The reason I really liked this idea, and wanted to share it, is because: (a) kids were motivated by it, (b) kids were forced to grapple with complex mathematical language, (c) kids got to play around (by drawing different graphs — a puzzle-y thing), and (d) kids discovered the Intermediate Value Theorem on their own.

Let’s think about the last point. The first 5 or so graphs students would draw would not satisfy the challenge. And they’d see the problem: that the graphs they were drawing were continuous. So the only way to satisfy the challenge would be to make their function discontinuous. So not only would they learn the IVT, but they’d really remember the restriction: you need a continuous function for the IVT to hold.

I’m sure many of you probably introduce the IVT this way. It’s certainly not new or revolutionary. But I am now excited to when I get to teach the IVT.

PS. I also am really impressed by this consequence (click link to see proof). The consequence of total boringness happens not to be boring at all!:

The theorem implies that on any great circle around the world, the temperature, pressure, elevation, carbon dioxide concentration, or any other similar quantity which varies continuously, there will always exist two antipodal points that share the same value for that variable.

Two Worksheets

ONE

On Thursday, I’m going to be introducing absolute value inequalities. Last year I used the picture below as motivation.

I then tried to work backwards to show kids absolute value inequalities. It wasn’t too hot a success. Certainly the “application” wasn’t a motivator, and working backwards just confused things.

This year, I’ve decided to start with a warm up. Without them knowing anything, I’m going to ask them to do this for the first 7 minutes of class with their partners.

I already can see the great questioning and discussion that this simple worksheet will generate between partners. And then, when we come together: WHAM! powerful! It’s a simple thing, but Oh! So! Delicious!

After that, after we see some patterns and make some conclusions… then, then I can throw up the picture of the bag, and talk about it meaningfully. And have kids work backwards from their own conclusions to finding a way to express that region mathematically, using absolute value inequalities.

TWO

I’m introducing limits tomorrow. I pretty much have carte blanche in what I do. Last year what I did was sad. Like SAD. Like: “Here’s what a limit is. Get it?” This year, I’m stealing pretty much from CalcDave wholesale. Here’s his calculus questionairre. And here’s what I made.

Pretty much the same thing. Then I’d like to somehow have them start thinking about how to get velocity from a position versus time graph. Haven’t quite figured that out yet. Either that, or Zeno.

A great Multivariable Calculus problem

Today I gave my multivariable calculus class a problem — a problem I give every year, that I found… somewhere. Maybe MIT, maybe an Exeter problem set, maybe a textbook. And if you ever want to see kids work together, and do some good problem solving, this is a prime problem for that.

Up to now, we’ve been working on vectors — and they learned vector basics (read: dot product and cross product). Here’s the question.

You have any tetrahedron. Sticking outwards from each face, orthogonal to each face, is a vector with magnitude equal to the area of face it is sticking out of. Prove that if you add these four vectors together, they sum to the zero vector.

It’s such a beautiful problem. I don’t have a totally geometric way to explain why this is true (we do lots of good vector algebra), but I do enjoy watching everything all come together. To me, it almost works like magic.

I then had a student ask an amazing extension question. (If they’re asking extension questions, you know it’s a rich problem.) He said: “Will this always work for any polyhedra? What about figures involving faces which aren’t triangles?” I, of course, decided I loved the problem. And I desperately want him to work on this for his final project.

I love the idea of this student taking this problem and seeing how far he can run with it. I mean: hello, gluing tetrahedrons together! (I expect him to make some stick models, if he does it!)