Hook, line, and sinker: Calculus bait

I was reading — as I think we all were — that New York Times article “Building a Better Teacher.” In that article, a number of ideas and sentences and thoughts leaped out at me, especially concerning Doug Lemov’s taxonomy. (Yes, like you, I’ve already pre-ordered the book and cannot wait for it to arrive.) One of Doug’s points is:

The J-Factor, No. 46, is a list of ways to inject a classroom with joy, from giving students nicknames to handing out vocabulary words in sealed envelopes to build suspense.

I love the idea of sealing things up and unveiling them. So in my calculus class, right after we finished anti-derivatives but before we embarked on integrals, I gave my kids 15 or 20 minutes and this picture.

I showed them a Chinese take out container which I shook (and it rattled), and I said it had very special prizes inside. I showed them a fancy envelope and gave them each a notecard that they would place in the envelope. With their name, and their area estimate.

Each kid worked individually — using anything they had on them like rulers, straightedges, calculators. One student asked if he could use a scale from the physics lab (I said no, mainly because of the time issue.) I did this in two classes. Both seemed into it, but one was definitely more into it than the other.

What was interesting to me was how hard it was for them. Not the estimating, or the making of triangles and rectangles and other smaller pieces. What was hard for them was being asked to do something that they didn’t know how to do. It happened multiple times that kids were sheepishly telling me that they didn’t know how to start (they had already drawn auxiliary lines and broke the figure up into smaller pieces — um… you DID start, darlin’), that they were doing it wrong (um, didn’t I say there was no wrong way to do this?), that they didn’t know the right way (um, see my last um). They were telling me this to assuage some part of their psyche that was telling them that they had to be right. I told them to STOP BEING CONCERNED ABOUT KNOWING THE RIGHT WAY and just TRY SOMETHING! Then they did.

I also mentioned that last year someone got the answer right to TWO decimal places — setting the bar high.[1]

At the end of the allotted time, I collected the notecards, put them in the envelope, and sealed it with a flourish.

I told them it would take a week or so before we could unveil the envelope (“but Mr. Shaaaaaaaaaaah”) and find out who came the closest to the real answer. And how would we find the real answer?

Calculus.

This was their hook for integrals. The next day (today) I introduced the idea of area under the curve being related to that anti-derivative thingamajig that they had been working on. I got at least 4 questions whining about needing to know who got the closest answer. I stoically responded “you’re going to find out when you figure out the true answer… soon.” The hook worked, and the bait is waiting to be won. For them, the bait is getting the surprise inside that dang Chinese take out box. For me, well, they are now curious.

[1] That was technically true, but slightly a lie. The exercise we did last year was different. I gave various pairs of students the same graph with different gridlines… and I had them estimate. So, for example, one pair of students got:

So clearly their estimation was going to be better — and it is unsurprising they could get an estimation to 2 decimal places. And last year we talked about how the more gridlines you have, the better your estimate can be.

When do we get to have fun?

Think Thank Thunk makes me want to throw my hands up in the air. I’m not a good writer, but that sentence was carefully crafted to be pregnant with ambiguity. Because with every post Think Thank Thunk author Shawn Cornally writes, I rejoyce… and I despair. Reading him is like reading Dan Meyer again for the first time (although they seem to have slightly different cause celebres, they actually are saying almost the same thing). It’s all obvious common-sense things. Motivate. Have the kids come up with the questions. Once the hook or need is there, pounce. Capitalize. It doesn’t have to be “real world.” It just has to somehow get the kids internally invested, not just by grades. With a question. And a need for an answer.

I feel inspired by what I could be doing, and like a total lame-oid for what I am doing.

Or as David Cox twittered:
dcox21 .@k8nowak Problem is, I never sucked until I met all you guys. Thanks “everybody.”

Yeah. Thanks guys.

Recently I’ve been inspired enough that I’m going to try to get some curriculum money from my school to spend time coming up with (short) activities to “hook” or motivate my kids for each of the major topics we cover in my classes. That’s not going to be easy.

Reading Shawn and Dan just underscore something I’ve been feeling all year. I mean, I’ve felt this to some degree every year, but uber acutely this year. I became a mathteacher because I wanted to impart that feeling of exhilaration and accomplishment to my students… to show them the beauty and applicability and serious-honest-to-god-creativity that is implicit in math work… to see doing math as fun — a million little puzzles all connecting in these random and unexpected ways.

Or more succinctly: I became a math teacher because I want my kids to experience the doing of math as inherently enjoyable. So I’m asking myself: when did I lose that as a goal in my work, replaced by the singular focus on understanding? Yeah, understanding is great, but that should only be the baseline of my teaching. My standards should be higher, and getting kids who don’t enjoy math to enjoy math (not just tolerate, or be able to do, but enjoy) should be the target.

I know, I’m already feeling sheepish now that all this is typed out. All my idealism is spilling out unfiltered. And tomorrow I’ll go back to the classroom and see that my Algebra II students still don’t know why \frac{x}{2} is the same as \frac{1}{2}x, and my calculus students still don’t know why x^{1/2}x^{3}=x^{7/2}, and I’ll remember why I have such a singular focus on understanding, jettisoning fun for more immediate concerns.

But that’s still probably not going to stop my brain to keep on going to the place it has been stuck all year… asking ad infinitum the question “when do we get to have fun?”

Gallons, Quarts, Pints, Cups

One of the kids I help after school showed me this drawing that his teacher showed him. After seeing him draw it in front of me, I know I will never forget the relationship between Gallons, Quarts, Pints, and Cups ever again.

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There was something powerful about watching him draw the picture for me, which is why I have the video, and not just an image.

When I googled it, there are a few good image hits (so it must be a common thing). I just had never seen it.

A Super Specific Multivariable Calculus Question

Hi all,

I have a question about multivariable calculus, that I need some help with. My kids and I are both slightly stumped about this.

The question we are asked — in a section thrillingly titled, replete with semicolon, “Parametric Surfaces; Surface Area” — is to find the surface area of “The portion of the sphere x^2+y^2+z^2=16 between the planes z=1 and z=2.”

In class, the formula we derived for surface area for any parametric surface \vec{p}(u,v) is

S=\underset{R}{\int\int} \left\Vert \frac {\partial\vec{p}}{\partial u}\times\frac{\partial \vec{p}}{\partial v}\right\Vert dA.

We solved this by converting the (top part) of the sphere to a parametric surface:

x=r\cos(\theta)
y=r\sin(\theta)
z=(16-r^2)^{1/2}

Then we defined \vec{p}=<r\cos(\theta),r\sin(\theta),(16-r^2)^{1/2}> (where \theta ranged between 0 and 2\pi and r ranged between \sqrt{12} and \sqrt{15}. (Those limits for r come from the fact that we want the surface area of the sphere between z=1 and z=2 — which correspond to r=\sqrt{15} and r=\sqrt{12} respectively.) [1]

So I calculate \frac{\partial \vec{p}}{\partial r}=<\cos \theta, \sin \theta, -r(16-r^2)^{-1/2}> and \frac{\partial \vec{p}}{\partial \theta}=<-r\sin \theta, r\cos \theta, 0>.

So to use our surface area formula above, we need to find \left\Vert \frac {\partial\vec{p}}{\partial r}\times\frac{\partial \vec{p}}{\partial \theta}\right\Vert. Calculating that out, we get it to equal \frac{4r}{\sqrt{16-r^2}}. Phew, now we have something we can plug into the surface area formula for that “norm of the cross product” thingie.

Here’s where the question comes in. We know

S=\underset{R}{\int\int} \left\Vert \frac {\partial\vec{p}}{\partial r}\times\frac{\partial \vec{p}}{\partial \theta}\right\Vert dA=\underset{R}{\int\int} \frac{4r}{\sqrt{16-r^2}} dA.

Why is it that when we finally evaluate this beast, dA is not equal to our normal area element for polar, namely r dr d\theta? For the answer to come out right, we need to let dA equal to simply d r d\theta.

WHY? Why don’t we plug in the normal polar area element?

Here’s my thinking. Even though we usually use dA to represent an area element, in this particular surface area formula, it doesn’t represent anything more than du dv (for whatever parametrization gets made). The reason I think this? When I look at the derivation of the formula, it defines du dv to be dA. Simple as that.

I used to think that dA had a fixed meaning: the area element in a particular coordinate system. However, I’m now thinking that it might mean different things in different equations? Either that or our book is being sloppy.

If anyone can follow what I’ve written here and has any help to proffer, I would be much obliged. It’s a small point — one that won’t really matter in the long run for this course — but both my kids and I would like to have this resolved once and for all.

[1] If you don’t see that, imagine you have this sphere and you make a slice at z=1 and another slice at z=2. You want the surface area of that little curved “ring” — and if you find the shadow of that ring on the x-y plane, you’ll get two concentric circles with radius \sqrt{12} and \sqrt{15}. That’s the region R that you will be integrating over.

Surprise ’em with what they don’t know

Sometimes it isn’t that we are bad teachers. And it isn’t that we aren’t giving students the lessons they need. It is that students aren’t willing to shore up their knowledge each night to make sure they know what they know, and figure out how to learn what they don’t know.

So I try to aperiodically remind them of that fact.

Yesterday, for example, I hinted to my students that they might have a pop quiz. We’ve been working on quadratics, and have seen questions like:

Solve 2x^2+5x+7=0

and

Graph x^2+10x-8=y

and the latest feather in our caps

Solve x^2+10 \leq 0

It’s a lot. And quadratic inequalities killed my kids last year. So I told my students to spend the night just reviewing the material and making sure that they can organize the information in their heads. They come to class today and I give them a two question pop quiz, both questions on quadratic inequalities. 6 minutes. Most are frantic. Clearly many didn’t shore up their knowledge.

I then tell them to stop and put their pencils down. I tell them it wasn’t for a grade. I tell them I’m not collecting it. They breathe a sign of relief. We then had a conversation.

What was hard about the pop quiz?
Did you think you knew the material?
Did taking this quiz demonstrate that? Or did it tell you something else?

It was a nice and short conversation and I think it really drove home the point: you think you know, but you have no idea.

So here’s something for you to consider doing, if you’re cruel like me: a very occasional fake pop quizzes can be a nice conversation starter about studying and nightly responsibility.

UPDATE: So in this case, the faux pop-quiz was only moderately successful. Last year so many kids didn’t know what to do on the 1D quadratic inequalities question on the final unit assessment.  This year they were less were confused. But still there were enough students who didn’t know how to solve it to give me pause. I realize now that we learned so many different types of linear/quadratic things that students kept confusing “what’s the question asking?” and “how do I solve that kind of problem?” So I need to come up with a way to emphasize at each point of the unit these two fundamental questions. And maybe designing a short activity where students are forced to answer those questions.

Favorite Tweets #2

I did this a while ago — posted my favorite tweet conversations from days past. Favorite, for me at the moment, doesn’t really mean “advice on teaching” but just random convos. The best kind.

…

k8nowak @samjshah aw. You like us! You really like us! *hugs*

samjshah @k8nowak naaah, *like* is maybe too strong a word. you guys just keep me slightly amused. paint drying < y’all < reality tv

calcdave @samjshah You do know “Glee” is not reality tv, right?

Fouss @samjshah With that bit of information, you’re kicked out of the club. Don’t let the door hit you on your way out.

…

dcox21 @samjshah thanks Sam. Re-mixed them to suit my needs, but borrowed the heck out of your stuff. I’ll send you the remix…for approval.;-)

dcox21 Thanks to @samjshah… http://img111.yfrog.com/i/vtkl.jpg/

dcox21 @calcdave Yeah, I wrote it as neatly as I could and then placed the kids in just the right place just before taking the pic. ;-)

…

mctownsley The Teacher Salary Project – teaching is easy…right? http://www.youtube.com/watch?v=czPRKh2ooOY&feature=player_embedded via @scsdmedia

…

SweenWSweens Not gonna lie: pretty sweet artwork, but I really wish you were paying attention to the related rates lesson. http://brizzly.com/pic/1EJC

…

k8nowak @JackieB hey for pi day we are planning a ‘family feud’ assembly, and i put on the survey ‘name the teacher that gives the best hugs’ and..

k8nowak @JackieB the frontrunner as well as the second and third highest vote-getters…English. no joke.

…

k8nowak Just gave a girl a hard time about dropping out of college while buying beer from her. TEACHER OF THE YEAR.

…

busynessgirl #needaredstamp “It will take me more ink to explain what’s wrong than you used to do this problem in the first place.”

…

samjshah cool! how to find things that nearly rhyme… if i were musical this would rock… http://www.b-rhymes.com/

calcdave @samjshah My name is Sam which almost rhymes with stamp. These words I amass to teach you all math.

samjshah @calcdave My name is Dave which rhymes with concave. Up in the schoolz, my kidz be amazed. With my crazy calc skillz, ya heard? Word.

…

sumidiot anybody know a source for this quote about lectures: info from notes of prof to notes of student without passing through mind of either?

…

samjshah um, how do i feel that justin bieber and luda did a song together? someone tell me because i just can’t figure it out. http://www.youtube.com/watch?v=kffacxfA7G4

sig225 @samjshah On behalf of all Canadians, I apologize for Justin Bieber. Please, take our Olympic Men’s Hockey gold medal …

…

jbrtva Guy I know asked me out for coffee…he wanted to tell me that we’re just friends. #didimisssomething?

k8nowak @jbrtva now you invite him for coffee, and inform him you’re not just friends, but mortal enemies. Advise him to sleep with one eye open.

samjshah @k8nowak @jbrtva invite him for coffee, and inform him that you’re not just friends, but in a serious, committed relationship. then propose.

k8nowak @samjshah @jbrtva invite him for coffee and inform you’re not just friends, but actually brother & sister. Expose family’s deepest secret.

samjshah @k8nowak @jbrtva invite him for coffee and inform him you are tyra banks in disguise. scold him for sending mixed signals. KISS MY FAT A**!

…

samjshah just bought my spring break tickets. SAN FRAN HERE I COME!!! and i am taking virgin atlantic one way there… TV!

dcox21 @samjshah stop by Porterville and I’ll buy you dinner. Best Mexican food north of the border.

samjshah @dcox21 that’s 4+ hours away? NO WAY! wait, did you say you’re buying…

samjshah @dcox21 @k8nowak what are all y’all talking about? i missed some bandwagon. there were probably cupcakes on it, i bet.

k8nowak @dcox21 SSHHHH. Do NOT tell him about the cupcakes!

dcox21 @k8nowak Hey if he’s not willing to take me up on a free dinner, I’m not saying nothin’ about the cupcakes.

Teacher Nightmare

I had a very realistic teacher nightmare last night. Not realistic in the sense that it could happen to me. But realistic in the sense that it was one of those vivid dreams where you feel emotion when having it.

The day of the final comes and it is in a strange building for some reason. Before the exam, I am on the phone with my mother talking about something important. Taxes, maybe. Then as the final exam hour approaches, I take my leave from my mother and go to the classroom and see my children milling about. I look for the exams but they are not in my bag.

Wait!

I forgot to photocopy them!

I start getting really flustered. Really flustered. This was a dream where I felt emotion, and I was literally freaking out. This is so unlike me. [1] I don’t know what to do. So I open my computer to print out a copy, and realized I never finished writing the exam.

I didn’t even have an exam!

What’s on my computer is a document with a test that I got from somewhere, but I meant to modify to fit my class and I didn’t. I couldn’t give that to my class. I look at last year’s final, and it covers totally different material for some reason.

I go to this teacher’s lounge, literally paralyzed because I don’t know what to do. I’m frantically seeing if I have time to fix up the final, but I realize I don’t. I keep thinking about my kids in the classroom wondering what’s going on. I keep trying to figure out what to do. But my mind is stuck at this point and isn’t working. I’m mentally paralyzed, stuck in my own special world of freak-out.

I continue to frantically try to fix things, but I can’t. Basically, this goes on for two hours. (The exam is supposed to last three hours.) I don’t go back to the classroom. I don’t even know if they’re waiting there.

That’s where I leave off the dream, and enter in the waking world. I was unprepared, and a terrible teacher, who just decided to burrow in a hole and hide because he couldn’t effectively deal with this challenge.

AWFUL.

Update: The next night I had another teacher nightmare. I was covering class for another teacher, and students were supposed to be taking a test. They weren’t being totally silent, so I made the edict “The next person who talks will get a 0 on their test.” And someone talked. And so I took their test and told them they had a 0. Then other kids were complaining about that, so I took their tests. And it became this horrible battle of wills, with the frustration level of both me and the kids rising quickly. Blah. I don’t have this problem. I’m not (consciously, anyway) scared of having this problem. Why am I dreaming about it?

[1] In real life, I always have everything taken care of — jots and tittles and all. I never go to bed without all my work done.