Author: samjshah

I’m not dead yet…

I’m sorry for being so incommunicado recently. There’s been a confluence of things that have prevented me from posting anything — and will probably affect me for a week or two coming up. Which is why I thought I’d at least say “hey, I’m not gone for good…”

1. Spring break just happened
2. The quarter is ending this Friday
3. I have to write narrative comments on every one of my students by next Thursday
4. I have to have finalized grades (meaning everything must be graded!) by this Friday
5. I’ve had two major computer catastrophes at the same time (one on PC, one on mac), both of which are thankfully now averted
6. We’re hiring in our department and this will take up some of my time in the next two weeks

If you take anything from this, I want to highlight #5 and say: BACKUP.

Seriously.

Backup. Don’t put it off. Just sign up for Dropbox (click this if you’re going to sign up and I get some extra free space!) or Box.net and make sure you don’t lose all those materials you’ve been creating/hoarding, and your gradebook doesn’t disappear into the electronic ether.

And if the title to the post didn’t make sense, just for fun:

GACK! Where’s that post?

I don’t know about you, but I have this huge problem. I do all this writing on this blog, and I do all this reading on other blogs, and I have this information overload. I will get to a topic I’m teaching and be like

“Shoot! I know someone out there posted an awesome mnemonic for the sum and difference of cubes. What IS it?”

or

“Wasn’t there a great way to introduce logarithms that I wanted to try?”

Immediately after that, unless I’m in a i-need-to-find-that mood, I forego looking for it and just recycle what I did last year. I read about all this good stuff, but I rarely use it in the classroom. It’s actually pretty sad, considering how much time I spend reading.

The problems:

  1. I subscribe to a ton of blogs
  2. I’d guess that less than 1/3 of the posts deal with “on the ground this can help you in the classroom here’s a worksheet or mnemonic” things
  3. I can’t really find the posts in google reader using a search. I just can never come up with the right words, because I always only have a vague recollection of what I’m looking for

So I wanted to create a virtual filing cabinet — with only posts that can help with on-the-ground teaching stuff.

Without further ado

My Virtual Filing Cabinet

(Or you can click on the link on the upper right hand corner.)

Three things.

1. If you want to know why I didn’t make this into a collaboration where a bunch of us would add to this list and it would become super comprehensive… I did think about it. The only problem is: if everyone were adding to it, it would suffer from the same problem as before. Too many things to click that I don’t want to wade through. My blogroll is like my virtual magazine subscription — and I just put on what’s useful for me. So to is this list my virtual filing cabinet, and it has to be useful for me. (Which is why is it centered around Algebra II and Calculus at the moment.)

However, I bet if everyone made something like this, it could be super useful. Because just like I find new blogs through other peoples’ blogrolls, I could find great things from other peoples’ virtual filing cabinets.

2. Does that mean I don’t want suggestions on things to add to it? Obvi I definitely want suggestions. If you have something that fits just throw it in the comments of the virtual filing cabinet page. I’d be much obliged.

3. If you care to know how I went about constructing it, logistically, it actually wasn’t hard. I use google reader, and I “star” my favorite posts. So I first went through our curricula and wrote down the topics I teach. Then I went back through my starred favorite posts and found which ones fit the bill (the criterion: anything that could be super useful on-the-ground in-the-classroom) and added ’em in the list.

All in all, it took about an hour to do it. Seriously.

I’m sure I missed a bunch, but I figured having an incomplete list is better than no list at all. (Though my retentiveness hates the fact that I missed a bunch. It’s also why I get stressed when I can’t read blog posts for a week, and can’t “mark all as read” because I might just miss the most best idea ever!)

I continue to “star” my favorite posts and every two weeks or so I add what I’ve found useful to me to this list. (All that takes me is about 5 minutes.)

Favorite Tweets #3

Here’s #1 and #2. It’s the start of my spring break and I’m about to head to San Francisco to meet up with some high school and college friends. Consequently I won’t have much to blog about for a while. I thought it might be nice to leave with a post of the funniest/best tweets of recent days. There have been a spate of good ones that I’ve been saving.

…

CardsChic If kids aren’t doing well on these outlines because I didn’t explain it clearly enough, is it bad of me to blanket add 5 pts to all scores?

samjshah @CardsChic sure – give ’em 5 pts. but before that, i would also tell my kids that i feel bad bc i feel i didn’t explain it clearly enough…

samjshah @CardsChic so… so… THEY’RE ALL GETTING A CAR. [massive applause] then be like “oh, wait, actually you’re getting 5 pts bc i messed up.”

dcox21 @samjshah that’s better than outtakes after a movie.

…

jbrtva Feeling pretty awesome that I made @samjshah‘s Favorite Tweets v.2 post. I must’ve gained funny points in the past couple weeks :-)

samjshah @jbrtva that was my favorite convo EVER. i could have continued that riff for … 10 to 20 years … actually, at least 10 to 20 years.

jbrtva @samjshah With my dating history, you might be able to! :-) That convo made me smile, too.

k8nowak @samjshah @jbrtva next thing you know Sam’s making “invite him for coffee and” t-shirts.

samjshah @k8nowak @jbrtva not a bad idea, not a bad idea at all.

CardsChic @k8nowak @samjshah @jbrtva As long as the back says #didImisssomething?

k8nowak @CardsChic @samjshah @jbrtva Genius. She stays.

CardsChic @k8nowak @samjshah @jbrtva I’m in! Yesssssssssssss! *pumps fist*

…

dcox21 @samjshah I don’t care if I made your list. I’m still not sharing the cupcakes. #eatingmexicanfoodalone

…

weemooseus @druinok Its bad when one’s significant other says, “Turn off the teacher!”

…

dcox21 Jabin requested a diaper change by telling me he had “junk in the trunk.” #fb

…

samjshah in other news, i got my “i only twitter with math teachers” shirt in the mail today. the gold is REALLY GOLD and the red looks awesome. A+.

samjshah for those who requested the narcissistic photoshoot with my twitter t-shirt: (1.) go here https://samjshah.com/2010/02/22/twitter-t-shirts/ (2.) scroll down (3.) drool

samjshah if you want to make my visage your wallpaper, i understand. i won’t stand in your way.

…

dcox21 No really, it’s ok to write things down. #needaredstamp

…

SweenWSweens Did you really think all you needed to do on that calculus question was use basic algebra? #needaredstamp

…

CmonMattTHINK @k8nowak @samjshah though I must say, while Im unfamiliar with Felicia Day, after visiting her site I have two words and theyre both “hubba”

…

dcox21 f(x) does not mean f times x. #needaredstamp

…

k8nowak So on the Family Feud pi day survey, I was the highest vote getter for both “meanest math teacher” and “nicest math teacher”

CardsChic @k8nowak Does this mean you win the Bipolar Teacher of the Year award? Shoot, thought I had that locked up…

…

Fouss http://twitpic.com/16sggu – Me and @amfago with our cool shirts on!

msgregson @Fouss that’s awesome!!!

Fouss @msgregson @cardschic the kids loved them! That’s all I heard about all day :)

…

k8nowak For those of you following along at home: IT guy casually mentions “his girlfriend” today.

…

dansmath @samjshah @k8nowak @JackieB @Fouss @dcox21 @msgregson @cnansen @RobertTalbert @ddmeyer – I HAVE a red stamp! http://img72.yfrog.com/i/e2f.jpg/

…

k8nowak @cwoody222 Today at Kohl’s I bought underpants from a former student. Fun times.

…

JackieB “How dare anyone think that you can transform a child if you can’t be transformed yourself” @ChrisLehmann #TEDxNYED

…

Stelladuma Dan Meyer makes me rethink.

…

CarissaJuneK hubby just realized that was jake gyllenhall…#stopstaringatrachelmcadams

…

SweenWSweens Here’s a good one… http://brizzly.com/pic/1NPJ

…

derekbruff been hovering just under 500 followers for a few days now. determined to hit 500. so: justin bieber.

…

Fouss @MSeiler try @k8nowak, @msgregson, @samjshah, @calcdave, @dcox21. That should be good for starters. :)

dcox21 @Fouss @mseiler @k8nowak @msgregson @samjshah @calcdave Batting fifth in quite a lineup. ;-)

samjshah @dcox21 first is the worst, second is the best, fifth is … expendable.

dcox21 @samjshah should I tell that to my children?

dcox21 @samjshah Hey wait! Did you just say I’m expendable?

Fouss @dcox21 @samjshah When it’s an all-star lineup, does order really matter? :)

samjshah @dcox21 hahahahaha. your fifth won’t even know what you’re saying. you’re safe. for now.

dcox21 @cannonsr got me. I should clarify…my contract /allows/ me to keep them all. @samjshah‘s logic makes it optional.

dcox21 @samjshah “I’m expendable? But Mr. Shah, look at me.” http://img521.yfrog.com/i/weib.jpg/

cannonsr @dcox21 The picture wins. Can’t trump the baby card.

samjshah @dcox21 omg, that’s WAY too cute. okay, you MUST keep him. but you are NOT allowed to give him sharp objects anymore. NO MORE.

…

Fouss It may just be the glasses, but Andrew Garcia (contestant on American Idol) reminds me of @samjshah. Anyone else?

…

jbrtva “What’s the worst part about math jokes? If you get them, you probably don’t have friends” HA! from komplexify.com/blog (via @samjshah buzz)

…

CmonMattTHINK When trying to plan interesting lessons or projects, I need to keep repeating to myself: The perfect is the enemy of the good.

…

k8nowak Steven has been silent all year. Today he raises his hand to answer a question. “Whoa, Steven talked!” “Oh yeah. I forgot I don’t do that.”

…

k8nowak My GR subscribers went up to 600 and then down to 599. First time I ever saw it go down. Knife to the heart, Internet. Knife to the heart.

samjshah @k8nowak don’t tell me you’re complaining about having 599 readers now, SHEESH. talk about DIVA.

k8nowak @samjshah I’m complaining about the instantaneous rate of change. I thought YOU of all people would understand. SHEESH.

samjshah @k8nowak df(t)/dt is negative. Aaaahahshahah.

…

CmonMattTHINK Somewhere out there in the ether there is a perfect analogy for trying to teach Calc to students who cant do Alg. I’ll come up w/ it someday

…

Fouss Verifying trig identities today!!! My FAVORITE day of the year!

calcdave @Fouss Haha. In my voice, that sentence would sound sarcastic.

…

jbrtva @samjshah @calcdave I propose we have a twitter brunch someday..maybe after we all meet at a conference or something. I nominate Sam’s place

k8nowak @jbrtva @samjshah @calcdave I second your nomination.

SweenWSweens @k8nowak @jbrtva @calcdave @samjshah Third. When’s the next big east coast math conference?

dcox21 @k8nowak @jbrtva @samjshah @calcdave You can all come to my house. August is great here. Ask Kate.

k8nowak @dcox21 @jbrtva @samjshah @calcdave it’s the only time to go. The manure odor is at its peak.

dcox21 @k8nowak @jbrtva @samjshah @calcdave Don’t worry, we’ll pen up the cows and pigs for y’all.

…

samjshah AHHHHH, i lost an hour. did anyone see it? when last seen, it was about 60 minutes. it was holding a lot of errands or some extra sleep.

h_math @samjshah Saw it, liked it so much I kept it. Came in handy.

macsmath Spring forward on pi day?! as a math guy, I feel gypped!!

…

busynessgirl If you have A=B+C, then when you take the log of both sides, it is log(A)=log(B+C) not log(B)+log(C) #needaredstamp

…

samjshah one of my students yesterday was using ROTXA and ROTYA to mean “reflect over the x/y axis.” LOVE IT! may start using it. ROTYA 4EVAH!

…

samjshah today when announcing the top AMC scores to the entire high school, i said “as we are basking in the glowing penumbra [of] pi day…”

…

k8nowak “Ms Nowak, I can’t measure in mm. This ruler only has in and cm on it.”

k8nowak “Oh. You’re right. Here, hold this while I jump out the window.”

…

k8nowak Harry: “I programmed my calculator to do law of cosines. Is that ok?” Matt: “YOU’RE A WIZARD, HARRY!”

k8nowak I wonder how long Matt’s been waiting to say that.

…

dcox21 @samjshah If I can get you to try this, there’s no hope for a carnivorous future. http://img257.yfrog.com/i/wvn.jpg/

samjshah @dcox21 nope, that looks gross. like a liver that someone cut an umbilical cord off of and threw sand on.

dcox21 @samjshah OH YEAH?! Well you just dangled a preposition. So, there!

samjshah @dcox21 what can i say? i live on the wild side when it comes to standard English grammatical practices. I spit on Strunk AND White.

dcox21 @samjshah and I bet you don’t give a &$&% about the Oxford Comma either.

samjshah @dcox21 I’ve seen those English dramas too. They’re cruel.

[note: http://bit.ly/dv3hvd%5D

…

lpudwell earlier this morning, i was in my office when i overheard the following….

lpudwell student: “professor, i have a question” colleague: “you only get one question per day, and you’ve already used your limit.”

lpudwell student: “oh, sorry.” (starts to walk away) colleague: “wait a minute, come back. i’ll let you ask yesterday’s question.”

…

samjshah best comment on an article on quantum mechanics: “I both understand & did not understand this article at the same time” http://www.nature.com/news/2010/100317/full/news.2010.130.html#comment-id-9736

…

FIN

Riemann Sums and Error

Sometimes I wonder about my sanity. Our school gets a 2 week spring break. I decided to teach my calculus classes on the last day before break. And I meet one of them during the last period. So yeah, sanity?

As you know, I’ve been working on Riemann Sums. After calculating them by hand [worksheet here], I had my kids enter this program in their graphing calculators.

We of course talked about why the program actually gives you the Riemann Sum. I’m going to expect them to be able to answer a question on the assessment about it. All that calculator stuffs was on Thursday.

They were coming to school on Friday with this program entered on their calculator.

I got home on Thursday at 9ish pm and when weighing the options of “show Stand and Deliver” or “create a lesson plan,” I just couldn’t let go of the fact that if I waited until after spring break to capitalize on this program, the momentum would be lost. So lesson plan it was. And I whipped the lesson plan [.doc here] below in about 90 or 100 minutes.

I told my kids that the fundamental question that we are tackling is: what is the relationship between the number of rectangles in our Riemann Sum and the error to the true value? And then we talked about what they already know … which is as the number of rectangles increase, the error decreases. So I modified our question: what is the MATHEMATICAL relationship between the number of rectangles in our Riemann Sum and the error to the true value?

We’re studying a a semi-circle of radius 2.

And so they use the program and come up with a bunch of data

And since they are interested in N and the error, they enter those values into lists in their calculator. Looking at a graph for low N values versus the error, they see this on their calculators:

So yes, they see that as N increases, the error goes down to zero. Mr. Shah’s eyes are wide open with awe.

That’s as far as we got in class on Friday. When we return from break we’re going to find a curve to fit this data. They’re going to try to fit this data to linear, exponential, logarithmic, and power functions using their calculators. It turns out that you can find a great power function to match this data. Seriously amazing. (The others don’t turn out well at all.)

That’s only for low N (from 0 to 20). Will our power function hit the N=500 point?

YEAAAAAH! This is where I started to get excited when I was creating my lesson. Because I thought that the error function would get really off so far out in the data. But heck, it’s pretty awesome that it hits at N=500.

I was curious HOW much our error function helps us with our estimation. So the last part of the activity is having students use only 75 Riemann Sums to estimate the area of the semi-circle. The error is shown below.

Not bad. But we have found a function models how much error using 75 rectangles will give us. So adding in that correction factor gives us a NEW estimation of the true area. And how much is this new estimation from the true value?

Um, using our error function as a correction factor gives us an answer that is 5 orders of magnitude better. Instead of an error of 10^{-2} we get an error of 10^{-7}.

I was really shocked and pleased by this! That’s where the lesson ends. Partly because I was too tired to make more, and partly because I want to move on in the course. I have a number of questions still lingering in my head, that I will be thinking about over break. Including why the error takes the form of a power function for this function (the semi-circle)?

Also, I still have a problem with the circularity of it all. I needed (a priori) the true area of the semi-circle to calculate the error for each N. Then we use these errors to find a curve to match the error associated with each N. This curve gives us a correction factor which gives us a better approximation to the true area. But if the point of all of this was to find the true area, we actually had to know it at the start to come up with the errors!

Some part of me wants to say that there is a good response to that. I suspect there is, and it deals with computing time. Like “say you want a good approximation. Start by assuming your true area is the Riemann Sum calculated for N=1000. Then use that to come up with the error curve. This error curve will then help you come up with a better approximation for the true area.” Or something like that. I don’t know, really. I’m just talking about nothing at the moment.

Riemann Sum Set Up

A while ago, I posted some of the quirks/concrete things that I’ve developed for my class that seem to work. I think those sorts of things are SO useful. And I love getting them from other teachers. In fact, someone posted about teaching distribution as THE CLAW and I was helping a middle schooler with that today and it was so helpful. So yeah, if you have some concrete things that you use to teach specific topics, blog about ’em or if you don’t have a blog, throw ’em in the comments below.

Currently I’m teaching Riemann Sums in Calculus, and I don’t teach it rigorously. My kids don’t need to use summation notation or anything. I’m focusing on the concept. So I have them do a few problems like this (with the picture) by hand:

Many of them struggle with three things: the left handed vs. right handed thing, finding the endpoints of each rectangle, and being able to calculate the Riemann Sum without drawing a picture of the function.

So I created a way for them to represent the Riemann Sum so that they (a) don’t mix up the Left Handed and the Right Handed sums, and (b) they can still sort of “see” the picture. It also helps them if the interval isn’t totally nice.

Before I show it, I want to say that it isn’t innovative or ground breaking. I almost expect people to say how stupid and obvious it is in the comments — or that everyone does something similar. But heck, I don’t care. It does help my kids who have trouble organize all the information.

Note, when you’re watching the video, the difference in how I set up the left and right handed sums…

Vodpod videos no longer available.
more about “Riemann Sum Set Up“, posted with vodpod

So that’s that… If you didn’t catch it, I put little tabbie things on either the left hand side or the right hand side of each rectangle base, to show which one we’re doing. That tabbie thing is there to remind students (a) that’s the side we’re looking at and (b) we’re concerned with the height of the rectangle. That’s also why I write it vertically, instead of horizontally. To show we’re talking height.

I also will probably use this setup when showing them functions that go below the x-axis. (And I will probably write the height of the rectangle under the rectangle base to highlight that the function itself is going below the x-axis.) And use that to parlay into a discussion of “signed areas.”

I can easily see this being extended in a more rigorous course to dividing the interval into n pieces. And discussions of where the most area is coming from, and what that means (e.g. when talking about velocity, that means an object traveled further in that period of time).

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?”