Guest Post: On Being Yourself While Doing Math

I got an email from Rebecca Zook, who is a fellow math blogger, who was partially inspired by a post from forever ago (“Don’t Judge a Book By…“), asking me if I wanted a guest post on the same theme. Well, I’ve never had a guest post, and this is a darn good one, and now is as good a time as any! So without any more fuss and muss…


On Being Yourself While Doing Math­

by Rebecca Zook

When I met my new math tutoring student and her mom wearing my celery green pantaloons and a dress that made me look like a fluffy yellow daffodil, I wasn’t sure how they would react.

They looked at each other.  They smiled with relief.  And then, they beamed at me.  Next we got down to business and had a massively productive tutoring session.

This experience caused me to reconsider my entire philosophy of teaching attire.

My whole life I’ve had my own distinct style, whether that meant wearing galoshes without socks regardless of the weather (preschool) or making a dress printed with the solar system from an old curtain I found at Goodwill (high school).

But I started my career as a math educator by teaching SAT math for a big corporate test prep company.  So, despite the fact that they hired me when I was wearing a homemade miniskirt printed with text from a French nursery rhyme and pictures of chickens, once I got into the classroom, I seriously curtailed my exuberance and dressed for my teacher-role.  My efforts to wear business casual mainly consisted of me wearing the same pair of black slacks almost every time I taught.

Later, when I left the corporate world and started my own math tutoring practice, I still felt the need to dress “professionally” when I began to meet one-on-one with my own clients.  I worried that if I really dressed like myself—instead of some idea of what a female math tutor should look like—students or parents would get turned off or distracted by my clothes.

But I started to ask myself, why was it “educational” to pretend to be less fabulous than I really am?  Why not wear a really awesome outfit to teach in instead of trying to “look normal”?

In that moment, walking towards my new student and her mom, what I experienced was nothing like I’d feared.  It was something totally different: appreciation, excitement, and even recognition.

What the heck was going on here?   Why were they so clearly excited about me looking so different than they expected?  They might just be relieved that I was a female math tutor that wasn’t afraid to be girly, or something.

Or, it might just be because my new immaculately student, who frequently came to tutoring sessions wearing a pristine corset paired with bloomers, high-heeled knee-high boots, and antique goggles perched on her head, just approved of my weird style.

But then I remembered some of my different students’ styles: the 18-year-old homeschooler with blond dreadlocks and a torso-length tattoo of a Buddhist goddess; the fifth grader wearing her private school uniform who fervently professed her love of Abercrombie; the seventh grader who wore classic rock t-shirts and cherished her florescent vintage sunglasses from the ‘80s.

I realized it was something completely different than my students liking my style.

It’s about being yourself while doing math.

I want to create a space where my students feel they can create their own solutions and find what’s best for them.   I want to help my students gain true confidence in who they are, whether that means how they dress or how they think.

When a student spontaneously makes up a song about even numbers to the tune of Michael Jackson’s “Beat It”, or leaps out of their chair to spin a certain number of degrees to solve an angle measurement problem, I feel I’ve succeeded in helping them be comfortable with themselves and asserting their own choices.

Maybe finding and embracing your own learning style isn’t that different from finding your own fashion style.

Maybe I’m crazy here, but when students only see one kind of person doing math, maybe they’re getting a message that they have to be a certain kind of person or dress a certain kind of way to be good at math.

I believe that you do not need to restrain your awesomeness/exuberance to kick butt at math.  In fact, the same passion, sense of adventure, and endurance that may lead someone to get a full torso tattoo of a deity can also serve them while mastering math problems.

You can learn math no matter what you look like, whether you’re into dressing like a daffodil, a Buddhist punk, or an Abercrombie fiend.  And you definitely don’t have to act, or look, boring.
BIO: Guest blogger Rebecca Zook is an online female math tutor who has been helping students get math into their brains for seven years.  Her blog, Triangle Suitcase, is about unpacking the process of learning in all its complexity, frustration, and delight.

MY THOUGHTS: If you’ve ever met me in real life (well, only two or three of you have), you know I am a lot of … something. And I love that about me. In high school, I was big into shopping at thrift stores and pairing together plaid golf pants with a pair of silver spray painted shoes with a ratty old tshirt. I was a clothing bricoleur. There are lots of reasons for that, but the consequence was that my sense of self was tied up in how I dressed. I saw myself as an unconventional almalgam, and my clothes were a conscious reflection of that. Well, because of that, I fully support students expressing that sense of self. And Rebecca has tied this same thing — this sense of individuality and choice and confidence — to how students do math. I like that. I also really was struck by this line:

Maybe I’m crazy here, but when students only see one kind of person doing math, maybe they’re getting a message that they have to be a certain kind of person or dress a certain kind of way to be good at math.

Holla! That gets to the crux of why diversity is important. It’s why we need female science teachers teaching AP courses, and female leaders in student goverment, and a diversity of ethnicities, religions, and sexual orientations in visible places in schools. You don’t need to be a guy to be great at math, and you can be gay and good at sports, and you can be a girl and in charge of student government. So yeah, it’s just clothing. But it’s kind of something more, if you look at it from a slightly different angle.

Favorite Tweets #4

Here’s #1 and #2 and #3.

msgregson @k8nowak facebook is suggesting that we be facebook friends … how does it know? wierd.

Fouss @msgregson @k8nowak Facebook is kinda scary sometimes…

k8nowak @msgregson @fouss we have a mutual friend @calcdave. I blame him.

calcdave @msgregson @Fouss @k8nowak Mwa ha ha! My evil plan is coming together.

samjshah @dcox21 Q: are you at the point where you are going to get a whistle like captain von trapp? “i’m incouragable!” “i don’t need a governess”

k8nowak lol @samjshah The @dcox211 children don’t play! They march!

k8nowak @samjshah @dcox21 21, even!

dcox21 @samjshah No whistle for me. I pick ’em off one at a time.

Fouss @msgregson I just wonder if there are other little groups of math teachers floating around. Or if we’re “it”.

msgregson @Fouss ohmygosh! i wondered the same thing!

calcdave @Fouss Well, most of the cool ones are here already it seems, eh?

Fouss @calcdave @msgregson True. They couldn’t be any cooler than we are. :)

JackieB Hey, I made it into @samjshah‘s favorite tweets post! Sorta. I quoted @chrislehmann. Alas, I guess my students are right – I’m not funny.

samjshah @JackieB that’s definitely the takeaway from this comprehensive and exhaustive list that i’ve painstakingly constructed. definitely.

k8nowak @samjshah FINE. I’ll order a SHIRT.

cannonsr I finally got to wear my Twitter t-shirt today! Thanks @samjshah. (Join the club @k8nowak!)

JackieB @k8nowak Can you get me one too while you’re at it?

k8nowak Matt & Scott just stopped by for a game of “one-dimensional infinite tic tac toe.” Matt: “It’s pretty much the worst game ever.”

CardsChic @misscalcul8 @k8nowak Speed dating game and they do all the work?? I’m listening…

misscalcul8 @CardsChic http://function-of-time.blogspot.com/2009/10/speed-dating.html

k8nowak @CardsChic http://function-of-time.blogspot.com/2009/10/speed-dating.html

misscalcul8 @k8nowak lol

k8nowak @misscalcul8 tweet collision!

CardsChic @misscalcul8 @k8nowak Haha! Thanks – I’ll have to add that to my list… and this is why I love Twitter. :)

misscalcul8 I feel like suh a bad teacher when I give out worksheets.

CmonMattTHINK One of my advisees audibly contemplated inducing vomiting so as to get out of going to her Precalc class.

CmonMattTHINK Don’t really want to think about how many students have considered that course of action for my classes…

MitchKeller An unnamed colleague: “If I were going to go back in time and kill someone, [Walter Rudin] would be at the top of my list.”

CmonMattTHINK @samjshah I feel the same about the similarly-named Royden, whose Real Analysis text I regularly referred to as “Hemorrhoyden.”

busynessgirl At the point where you accidentally “grade” the answer key, it’s time to go home.

BuiltOnFacts When the department email begins with “Here is an update on the odor in the building”, I am not optimistic.

dcox21 @samjshah I’m really liking you for putting all those great resources in one place. I forgive you for not taking me up on the dinner offer.

samjshah @dcox21 and i thought you liked me for my dashing good looks and quirky sense of humor. you are SO superficial.

dcox21 @samjshah That too. 140 characters, you know.

calcdave Somedays it just feels like this: http://bit.ly/aaAsRV

k8nowak For a while there I thought it was Thursday then I realized it’s Wednesday but we have Friday off anyway so it’s all good.

calcdave @k8nowak :-| :-( :-D

k8nowak If one more kid comes up and says “Calculator?” with out any additional words, or a “please,” I’m going to shove one up his butt.

misscalcul8 @k8nowak You didn’t wanna say butt, did you?

k8nowak @misscalcul8 Rage. I have it.

revrev “Oh yeah? If I die my ghost is going to haunt you!” “Yeah? Well your ghost is going to see some disgusting stuff.”

BuiltOnFacts Last famous physicist/writer to visit campus: Roger Penrose. Got his autograph on his book. Next week: S. Hawking. Autograph less likely.

samjshah @k8nowak http://brizzly.com/pic/1Y7D and i definitely think i should get points for saying “peggy hill”

madcaptenor Human beings bond by talking shit about other human beings. This is why I don’t like human beings, and identify as an icosahedron.

samjshah okay just two more comments to write tonight. both are really hard kids to write about… tricky situations…

k8nowak @samjshah “Works well when under constant supervision and cornered like a rat in a trap.”

k8nowak @samjshah “This young lady has delusions of adequacy.”

calcdave How do you solve an integral like Maria?

CmonMattTHINK @calcdave I’m guessing with inverse tangents. http://en.wikipedia.org/wiki/Witch_of_Agnesi

CardsChic Me: I’m not touching that with a 10 foot pole. Boy: How about a 9 foot one? #ILoveMyStudents

JackieB If you share your deepest fears about yourself as a teacher with someone, and they start laughing, does that mean your fears are unfounded?

msgregson My favorite users are @misscalcul8 @Fouss @samjshah ! Perhaps this is love ;) http://twtlab.com/love/

samjshah @msgregson perhaps this IS love… like the “mmm cheesy fries” love, though, not the “put a ring on it” love

msgregson @samjshah how did you know that cheese fries are my favorite? you really know the way to a girls’ heart.

samjshah @msgregson then when you come to NYC, we have to go to a place i discovered this weekend, with the best OOEY GOOEY CHEESY TOTS!!!!!!!

msgregson @samjshah i like how you said “when you come”, not “if you come” … it IS love. :) and, yes, it’s a date, when i come!

park_star my students have started making fun of one another by calling each other “cylons.” i heart nerds.

pepepacha Good Idea: Having my 8th graders prepare a math fair for our elementary students to distract them from premature “senioritis”

CmonMattTHINK If f & g are both increasing on a given interval, then so is fg. If true, prove it; if false, give a counterexample. (LOVE q’s like this)

ddmeyer “Death march to calculus” is my favorite new expression. #ncsm

ddmeyer What’s your #NCTM day 1 scorecard? I posted 3 walkouts and 2 headbangers that tossed my grey matter like a spinach salad. Overall, a win.

k8nowak @JackieB Normally I can’t help but make my sarcastic comments out loud. I make a lot of friends at conferences.

jbrtva @k8nowak @JackieB I, too, love making snarky comments at conferences to make friends with the people around me. :-)

k8nowak @JackieB @jimwysocki @jbrtva To clarify, by “make a lot of friends,” I meant “totally alienate everyone within earshot.”

k8nowak @JackieB @jimwysocki @jbrtva But hey, if you don’t have anything nice to say, sit next to me!

cannonsr @JackieB @k8nowak et al. I write the snarks in my notes. And pass to share with the neighbors.

jimwysocki Got a take-away from my first session. Challenge students to bring in problems for me, demonstrate struggling.

jreulbach @Fouss they will – too much! Better review your calm down quote. I say 3-2-1 they say math is fun.

samjshah i’m jealous, all. can we plan to go to a conference together next yr? maybe we should just plan our OWN small conference abt blogging?

samjshah that last tweet was for everyone at NCTM and everyone else who is JEALOUS of them. im kinda serious about hosting a conference. NYC?

k8nowak @samjshah How about next Wednesday?! j/k. I think some prior coordination to invade a conference would be nice.

SweenWSweens @samjshah I’m in, got such an idea in October, but was waiting for more street cred to suggest it.

SweenWSweens @samjshah @k8nowak I see Kate’s idea and raise October Baltimore NCTM. Also, NYC is prolly too far north for them VA and NC peeps.

CmonMattTHINK @samjshah @k8nowak @SweenWSweens I can just see submitting the PD $ proposal… “Location: various bars in NYC…”

dcox21 @samjshah Thinking the same thing. Think I’d get more from talking to people I already know I respect than from rolling dice on conference.

mctownsley @dcox21 @samjshah what’s the name of this math/blogging/twitter/teaching crew? seems like a good start for a mini-conference

samjshah @dcox21 @CmonMattTHINK @k8nowak @SweenWSweens i know the good ones, just so you know. the right mix of dive and quality craft beers.

SweenWSweens @samjshah @dcox21 @CmonMattThink @k8nowak There’s gotta be bars in NYC that sound like they could be conference venues.

samjshah 1. go to NCTM conference, 2. meetup at nyc bars, 3. hold our own miniconference (designed to be useful), 4. do nothing and let the idea die

cannonsr @SweenWSweens Like the bars called “The library” near university campuses. Totally

jreulbach @samjshah YOu had me at NYC bars…

k8nowak @samjshah et. al. An unconference. Like BarCamp.

jreulbach @k8nowak Bar camp?

samjshah @k8nowak et al. yes. our own format. about interactivity, convos, playing with the web. less about leaders and more about those coming.

k8nowak @jreulbach http://lmgtfy.com/?q=barcamp

samjshah @k8nowak wow, i’ve seen that *in theory* before (see this cool site), but never used in practice. OH SNAP!!! @jreulbach got served!

SweenWSweens @samjshah Sounds like those venues are the start of a sweet summer 2010 “conference” where we could brainstorm about any future ones.

k8nowak @samjshah @jreulbach I feel kind of bad for picking on the noob. Sorry. I’ve been waiting for an excuse to use lmgtfy.

k8nowak @SweenWSweens @samjshah Yes. This calls for a preconference planning conference.

jreulbach I am in for summer. No kiddo or new job conflicts…

jreulbach @samjshah Can’t believe you said OH SNAP!! LMAO.

SweenWSweens @samjshah @k8nowak Any1 else close ? The Internet is so weird, everyone is so spread out. How do the tubes bring things so from so far away?

k8nowak @SweenWSweens Well, see, it’s a SERIES of tubes.

k8nowak @SweenWSweens @samjshah I don’t know who else is near-ish, but we could announce on blogs and see who bites.

CmonMattTHINK @SweenWSweens @samjshah @k8nowak Yeah how does there not exist a Google Map or something with all of our locations on it?

samjshah @k8nowak @CmonMattTHINK here’s our map! enter your info! (unless you tweeted about a map when i was making one) http://maps.google.com/maps/ms?hq=http:%2F%2Fmaps.google.com%2Fhelp%2Fmaps%2Fdirections%2Fbiking%2Fmapplet.kml&ie=UTF8&lci=bike&dirflg=b&hl=en&msa=0&msid=102961433061679876855.000484f12a77524f148fe&ll=40.670125,-73.955283&spn=0.00721,0.01929&z=16

SweenWSweens @samjshah Ugh haytchu, I was literally typing the one I had just made in. You win this round! But watch your back!

samjshah @SweenWSweens *pffffft*

cannonsr @samjshah Loving how this map has 63 views within 10 minutes.

SweenWSweens @samjshah @k8nowak I’m not sure posting on my blog will help, as seen in figure 1-a http://mobile.tweetphoto.com/19648985

dcox21 @sweenwsweens @k8nowak @samjshah If you look reeeeeal close… #photocaps http://twitpic.com/1hwwir

samjshah ha! my dad sometimes sends me random links he thinks i might find interesting. today he sent me a link to @ddmeyer‘s blog. HA!

ddmeyer @samjshah My folks do that too. Can’t decide whether to be insulted or worried.

SweenWSweens @samjshah next week your mom will send you a link to my blog!

samjshah @SweenWSweens @cannonsr i feel there is a good “yo mama” joke in all of this. i just am not all that excited to hear it.

ddmeyer Becker. Developing computation. 6F. “Take one of each.” Okay. #nctm10 http://yfrog.com/5m4mswj

k8nowak @ddmeyer Wow. I hope you brought the big backpack.

ddmeyer @k8nowak Seriously. And check this: he just ran out of handouts and has offered to send them out to latecomers … BY US POSTAL MAIL!

dcox21 @ddmeyer @k8nowak What’d he make those handouts with a typewriter?

dcox21 @ddmeyer @k8nowak Upside: at least he didn’t use Comic Sans

dcox21 @ddmeyer If I caught the sacrasm, you had enough. Note to tweeps: if Dan starts out with “Hey, guys,” it’s gonna be sarcastic.

ddmeyer Hey you guys! Has anybody heard anything about this “Singapore Math”?! #nctm10

ddmeyer Obvs. I didn’t pour enough sarcasm onto my last tweet. I’m drinking a shot of Wild Turkey whenever a presenter refers to “Singapore Math.”

dcox21 @JackieB @cassyt @jimwysocki @ddmeyer Hug? What was this, an English teachers conference?

CardsChic @dcox21 @JackieB Wait… so since I teach English for 1/8 of my day, is it okay that some of my students want to hug me sometimes? ;)

JackieB @dcox21 Yeah, I admit it. I hugged at NCTM.

MEAN (grrr) value of a function!

In calculus today I was talking about how to find the average height of a function. Some kids just have a hard time understanding the concept. I always show them a few functions on certain intervals and I ask them what they think the average height would be. Just to initially test their intuition on the concept.

Some see it, and understand it; some don’t. All certainly have trouble articulating why they chose that value.

So I have two things that work for me, when explaining this. There’s some handwaving, but the focus is on the idea, and building intuition.

The first thing is we talk about how we would approximate the average temperature somewhere:

We take a bunch of temperature readings, and we add them together and divide by the number of readings.

How do you make it more accurate?

MORE READINGS!

How do you make it more accurate?

INFINITY OF READINGS!

What helps us deal with infinities and infinitessimals?

CALCULUS!

So that’s how we get started.

Then when I want them to understand the formula — f_{avg}=\frac{\int_a^b f(x)dx}{b-a} — I give them a little dumb, cute story.

So an EVIL mathematician has an almost 2 dimensional fish tank. Really thin. Sad for the fish. Which are almost 2 D. And the mathematician likes to lay a strip of plastic on top of the water,  and constrain the fish in these weird shapes.

(In this case, the mathematician is constraining the fish in an x^2 from [0,1].)

You come along and want to GIVE THE FISH WHAT THEY WANT: a normal rectangular water to swim in.

So you yank the plastic strip away, and what happens to the water?

IT ALL LEVELS OUT!

What shape does it make?

A RECTANGLE!

Does the amount of water change?

NO!

What’s the height of the rectangle?

THE AVERAGE HEIGHT OF THE FUNCTION!

So by then, we have on the board:

And since the amount of water didn’t change, they know that the area of the red rectangle and the area of the blue rectangle are the same.

That makes sense to them.

I then threw this up and almost all of ’em got it!

So that’s my way of building their intuition when it comes to average height of a function. It’s not like it’s hard for them to apply the formula, but I think this little thing makes it more conceptually manageable. And if they forget the formula, they can just do the “fish tank problem.”

Solutions to parabola problem

I’m going to post the way I worked out the two recent problems that I posted. Today I’m going to focus on the second problem first.  Other people had different solutions they threw in the comments to the original posts, so you can look there too.

FIRST PROBLEM:

Statement: A particle is moving along the curve y=x^2-x at a constant speed of 2\sqrt{10}. When it reaches the point (2,2), you know \frac{dx}{dt}>0. Find the value of \frac{dy}{dt} at that point.

Solution: I imagined the particle moving along the curve, and it being played on a film. The particle follows this path and is going at a constant speed:

So then I said: we only care about the particle around (2,2), so I mentally zoomed in near that point:

So we don’t care about the rest of the picture. The particle is actually moving in a straight line in the area we care about, and this line is y=3x-4 (we found the equation of the line tangent to the original curve at the point (2,2)). So this greatly simplifies how I had to think about the problem. Where we care about things, the particle is moving in a straight line at a rate of 2\sqrt{10}.

So then I thought about the velocity vector for the particle — moving in the direction of the line at a rate of 2\sqrt{10}. And this vector is composed of the velocity in the x-direction and the velocity in the y-direction: \frac{dx}{dt} and \frac{dy}{dt}.

And we just have to use the last piece of information that we haven’t used… That the ratio of height/length of this triangle is 3 (the slope of the line — the direction of velocity — is 3). So we can solve this with a bunch of different ways, but I found the easiest to just make similar triangles and solve:

(I calculated that the hypotenuse of the second triangle was \sqrt{10}.) Clearly we can deduce that dy/dt=6.

In essence, this is the exact same method that other people used to solve it, but it took me to actually zoom in and picture what was going on with the particle to figure this problem out so that I conceptually had mastered it.

The way I approached the second problem comes later.

Parametrization, Parabolas, Calculus, OH MY!

Okay, so a second problem in a row! This one is a straight up calculus one, from the 2008 AP Calculus BC exam — multiple choice section. The teacher of that class asked me if I could work this problem — and I admit I struggled. She showed me her solution, and then I left thinking “it couldn’t be that hard…”

When trying to fall asleep today, I started thinking of it and I was able to solve it in a different way.

Without any more preamble, if you care to try your hand at this:

A particle is moving along the curve y=x^2-x at a constant speed of 2\sqrt{10}. When it reaches the point (2,2), you know \frac{dx}{dt}>0. Find the value of \frac{dy}{dt} at that point.

As usual, feel free to throw your thoughts, solutions, etc. in the comments below, if you want. I bet for many of you this will be super easy, but for the few of you who struggle through it (sigh) like me, you might find it actually frustratingly enjoyable.

Oh, and also throw down there if you get stuck and care to see my solution… It’ll motivate me to actually type it up in a timely fashion.

A good problem solving problem

So… I am in this “problem solving” group at school, and we spent today trying to come up with a lesson centered around problem solving that we could use for one of our classes.

I’ve been really hankering to make one of these hyperboloids out of skewers:

and I thought it would be a great investigation for my multivariable class to figure out if indeed that was a hyperboloid of one sheet. I figured it would take a number of days — at least one to create one of our own, and a good number to figure out how in the world we would come up with the equation to define that beast. [1]

Of course one of the things we talked about in our problem solving group is how to bring the questions down to simpler questions — and then generalize. So I immediately thought of these drawings I spent hours of my childhood making:

[Yes, clearly my mother was happy that I found these to amuse myself with, instead of whiiiiiining “I’m so BORED… we have NOTHING to do in this house” as I did way too often.]

If you look, they define a really nice gently sloping curve.

So my question is: what is the equation (written in terms of x and y) for the curve above?

The first segment goes from (0,5) to (0,0). Then another segment might go from (0,4) to (0,1). Another segment might go from (0,3.5) to (0,1.5). (So however much down you go on the y-axis, you go that much right on the x-axis.)

I haven’t solved the harder 3-d question yet, but I had a heck of a time solving this 2-d question.

Since I had so much fun, I thought I’d share the problem with you!

I’ll post my solution later, but if you want to throw your solution down in the comments and how you came up with it (or blog about it), awesome. Just like with this “circles, circles everywhere” problem where someone posted the most elegant solution EVAR.

Fourth Quarter in Multivariable Calculus

We’ve just started the fourth quarter of school, and I’m starting to see some of my kids slip away from me, getting farther and farther out to sea — little bobbing dots in the distance.

In multivariable calculus, I’ve designed the course to prevent that from happening. I have a serious fourth quarter project for them — totally designed and executed by them. (They have to make a prospectus, they come up with a concrete and reasonable timeline, they troubleshoot problems that arise, and they even come up with the grading rubric.)

But I do something else to change things up, which actually is pretty neat.

We watch videos.

These multivariable kids will soon be off to college and will likely take tough, lecture-based math classes, where you don’t get the individualized attention you get in high school. My kids have never been exposed to college lectures, have never learned to taking solid math lecture notes, and have never learned to work through the difficulties that the lectures might pose.

So in the fourth quarter, I teach about 1/3 of the final unit… and then I hand the class over to Denis Auroux of MIT. Yup, I download a bunch of MIT OpenCourseWare 18.02 lectures and we watch them together in class.

Today, we watched Lecture 19 on vector fields and line integrals in the plane.

After we get to harder lectures, we’ll get in the habit of spending a day watching a lecture, followed by spending a day going over questions, tying the lecture to the book, and doing problems.

I feel like part of me is lying to ’em, though, because Denis Auroux is such a clear expositor with amazing board technique. Not standard, by any means. (But why burst their bubble now?)

If you were teaching an AP Calculus course, I would definitely have my kids watch an 18.01 (single variable calculus) lecture at least a few times after the AP test.

PS. I should really show them something like this — which I got exposed two a few times as an undergrad. Not horrible, but you really have to be focused.