How Close is Close Enough?

For my 8th graders, homework for day 2 consisted of a worksheet where students determined which set(s) included given numbers. Pretty easy stuff. But I threw one of my favorite problems at 'em to see what they'd do with it.

What's the sum of 1/2 + 1/4 + 1/8...?

At first they're thinking, "not possible 'cause it goes on forever." I told them to try it anyway.

The two most popular answers were ".99..." and "1." But those who answered "1" were quick to admit that they just rounded off. We open the discussion and I was very pleased with how thoughtful and respectful everyone was. These kids were really interested in getting to the bottom of this. It was a great opportunity to demonstrate that often times drawing a picture will allow you to see things in a problem that you may not otherwise catch.

So we draw a square on the board and shade 1/2. Then we shade 1/4, then 1/8 and so on. They soon see that the square will eventually be full.

Me: "So is it 1 or is it just really close?"

"Really close. Because the square is never completely full. You always have half of the remaining area that is unshaded."

Good. So let's see how they handle this.

"Alright, what's 1/3 as a decimal?"

".333..."

"Ok, and how about 2/3?"

".666..."

"what's 1/3 + 2/3?"

"1."

"And what's .333...+ .666..."

".999..."

"So does .999... = 1 or is it just really close?"

At this point they admit that it looks like it's equal but it just doesn't make sense. Time to to talk about what it means to be infinitely close to something. This is always a fascinating discussion.  We discussed the idea of a neighborhood and how if .999... does not equal 1, then there must be a number between them. 

"Give me the number and I'll shut up", I tell them. 

One kid says," How about .0 with a repetend, then a 1?"

But another student catches this, "If the zero goes forever, when do we add the 1?"

 It amazes me how these kids can grapple with the real "stuff" that is mathematics.  These same questions that got me hooked as I was taking my analysis classes in college are finding their way into the minds of 8th graders.  And you know what?  They get it...at least as much as they possibly can. 

Man, I love this job!

Is This Wrong?

In a 6-1 vote, the Los Angeles City Board of Education decided to turn over 250 of its schools over to charter and other private operators.  I'll definitely watch this story unfold with tremendous interest.  Although I don't really understand all the ramifications of such a decision.  I have to ask: Is This Wrong?

Is turning schools to charter groups and/or private operators going to foster competition and if so, is that a bad thing?

Will this help turn teaching into a profession where innovation is rewarded?

Do the teachers' unions actually have students' best interests at heart?

Who stands to profit from this? And do these folks care about education as much as they care about making money?

Some have said that this is a direct result of high stakes testing and the one-size-fits-all philosophy of education that inevitibly Leaves Children Behind. 

But were we doing such a good job before NCLB and the high stakes test?

I don't know, but I'm curious.

Teacher of the Year: Stevie Wonder

When I was about 12 years old, my parents took my brother and I to see Stevie Wonder in concert.  It was my first real concert experience and most of it is now a blur.  But 25 years later, the one thing that sticks out in my mind turns out to be something that had nothing to do with Stevie Wonder's music.  It had to do with a lady in the front row who couldn't carry a tune if it were strapped to her back. 

About half way through the concert, Stevie (or is it Mr. Wonder?) interacts with the crowd and decides to hold a singing contest.  He gets three volunteers from the crowd and they each get a turn singing Satisfaction by the Rolling Stones.  The winner gets to sing the song of his choice with Stevie Wonder himself.  Talk about the opportunity of a lifetime. 

Two of the three could sing very well, but I don't really remember much about their performances.  However, the third contestant was very sharp.  So much so, that even my untrained ear could tell that this lady couldn't sing.  Here is the impressive part:  in the middle of her singing, Stevie Wonder stops the band and has them adjust the key of the music to fit her voice.  He recognized the exact key in which she was singing and made the adjustment to fit her.  It did't work--she still stunk, but that isn't the point. 

As teachers, we need to do the exact same thing every day.  No matter how well we construct a lesson, we need to be ready to adjust to the kid who continues to sing off key.  You can't plan for that.  The band didn't practice the song in every possible key just in case they had someone who couldn't sing with them.  They knew their song, they understood the progression and understood what to do if they started somewhere different than where they had planned. 

I believe that an effective teacher is going to be the one who can recognize where a student is in relation to where the objective is, meet him where he is and adjust the plan accordingly.  It's not so much about having a great engaging plan all the time.  We can plan a symphonic lesson plan in which all the small parts fit together into a wonderful investigation or lecture.  But it's what we do when the kid playing the oboe doesn't hear what everyone else hears and plays the wrong notes that really matters.

Dear Sam,

Just read your post about being a fraud and you hit the nail on the head.  But you may not have hit the nail you were aiming at.  You a fraud?  Come on!  I have never met you but I can hear your voice with every sentence you write.  You can't fake that.

You know what I'd expect if I came to see you teach?  I'd expect to see a guy who cares about his students both in and out of class.  I'd expect to see a teacher who does his best to reach his students where they are, lift them to where they need to be and encourage them to become what they could be.  I'm pretty sure Socrates didn't have lesson plans and we're still talking about him.  All he did was ask questions. I'm pretty sure you do that too. 

Don't worry about putting your best stuff out there for all of us to see.  We all clean up the house when we have company over.  Every time you post,  you are inviting us to your classroom.  Thanks for that. 

So before you beat yourself up about not being where you want to be, remember this: None of us are!  That's the nail you hit.  We all keep striving in this game and none of us has completely figured it out; we merely get glimpses of what could be.  And anyone who tells you otherwise is the real fraud.  It took me 14 years of teaching to get to where I am and I still make rookie mistakes, have lessons that flop, get irritated with kids who won't engage and still don't exactly know what to do with kids who are bored.  

So thanks for the honesty, but sometimes we get tangled up in the accidents and forget the essence.  At your essence, you're a teacher; plain and simple. 

Best to you this year. 

David

p.s. Now if I find out that you're not really a teacher and all you did was stay at a Holiday Inn Express last night, then I'm gonna be pissed.

The Evolution of the Mathcast

Two years ago my principal approached me about getting a SmartBoard for each of the math teachers in my department.  I had no idea what he was talking about.  Heck, before I came here, my daily tech decision was: Vis-a-Vis or Expo?  So when he starts talking about this board that lets you interact with the computer whose screen is projected back on the board, I think my head almost exploded.  Didn't have a clue how I'd use it.  Against my better judgement, he ordered them anyway.  They came in a couple of weeks before school started, I helped him install one on my wall and off I went.  No training, no direction.  Just me and my computer. 

I particularly appreciated the ease by which I could do the drawings and graphs which were such a drag with an overhead.  I hated to give up my vintage set of chalk board drawing tools, but it had to happen.  A guy's gotta grow up some time.  As I was toying around with some of the features, I noticed that there was what looked like a record button.  We played around with the thing and figured out how to record the annotations but had a tough time getting the sound to record.  I played with the settings and in walks a wireless lapel mic.  Got it up and running and away I went. 

At first the recordings were a train wreck (I'm still not completely pleased with the quality of some of the examples).  The default file was .avi and I would post the lessons through the school website.  The problem was a student would click the link, go make a sandwich, do the dishes and come back just in time for the recording to play.  One of our district IT guys suggested I compress the .avi in MovieMaker to allow it to open faster.  Keep in mind, I really had no idea what I was doing.  I was basically swinging at pitches in the dirt hoping to connect.  Anyway, I had my first examples online ready to view by Christmas. 

Man, we thought we were cutting edge.  Then last year, I stumbled upon Tim Fahlberg's wiki. Turns out this guy has been doing mathcasts since I was in high school.  He practically invented this stuff.  I contacted Tim and he turned me on to Camtasia Studio.  It's a bit pricey, but it allows one to render the videos in many different formats.  It's also very easy to use.  Eventually this led to a channel on blip.tv as well as a podcast through iTunes.

The learning curve has been pretty steep but has started to level off a bit.  I am now passed the point of wondering how much can be done with this technology, but now I am wondering how to best use the ability to create dynamic notes for my students.  The nice thing about the channel on blip.tv as well as being on iTunes is that kids can subscribe and download the examples so they can take them wherever they go.  It's amazing to think that I have kids watching these things on their iPhone's and PSP's.  But they are.  So the question is how do we best use this tool?  Chris Lehmann suggests inversions.  What say you?

What to do...

I have a bit of a dilemma.  Three years ago I was brought to a GATE magnet school to teach a bunch of advanced kids. 

The thought was, "Rather than ship the really smart kids to high school to take the geometry class, we'll just bring the high school teacher to them." 

I didn't mind this idea.  In fact, it quickly began to grow on me.  I was teaching precalc, algebra 2 and algebra 1 at the high school and figured the change might do me some good.  The tough part was that I had no idea what to expect.  I was going from 55 minute periods to 94 minute daily blocks. 

What the heck am I going to do with a group of middle schoolers for 94 minutes? 

So after many discussions with my principal, we decided that since the 94 minute block was intended to allow for grade level instruction as well as time for intervention,  I'd just use my extra time for enrichment.  But what do you do to enrich them?  Do you accelerate the students so that they can be ready for algebra 2 as freshmen?  Do you spend the extra time doing all the cool stuff that no one else has time to do because they are worried about pacing guides and benchmarks?  We've decided to go ahead and accelerate them.  I'm still a bit unsettled about it, though.    Is it really that important for an 8th grader to complete geometry so she can take algebra 2 in 9th grade?  The kid is on pace to take calculus as a junior.  Is that better?  At this point, I don't know.  In the era of "the TEST", it seems that as long as I have the data to back up what I am doing, it doesn't matter what we choose.  My kids have the test scores, but I am not convinced that what I am doing is the right thing. 

I am definitely in unchartered waters.  As a high school teacher, I was simply one of many.  My last year there, we had a pacing guide and all common assessments.  It was pretty lock step and I was miserable--loved the staff and the students--but hated the system.  Now I have a bunch of autonomy because I can keep the powers that be off my back with some good scores, but I am not sure what direction to go.

Last year, the decision was a bit easier because I got to hand pick the students who would move into the geometry class. I taught both advanced 7th grade classes so I knew they were ready.   This year--not so much.  I have two classes made up of kids from two different teachers.  Some are ready to be accelerated and others look like they are going to need to spend some serious time with the algebra.  My question for you all is this:  how do you plan for these classes?  Half of my students have already been through algebra, a quarter of them have been exposed to it and the rest look like they may not know what a variable is.  I've never been here before, so I don't know what to expect.  If you've ever considered commenting on a post here, now's the time.  Hit me up.

Whatever It Takes

Our campus has been having some great conversations centered on developing tiered lessons that allow for differentiation depending not only on ability level but on learning modality.  How can we reach a student at their appropriate cognitive level while respecting whether they are an auditory, visual or kinesthetic learner?  Now, I am no cognitive scientist nor am I an expert at developing curriculum.  But I do know that meeting kids where they are is a good idea.  How we implement that is a different story.  Haven't figured that one out.

Last year I had a student who would have bounced off the walls if I didn't keep him engaged.  Getting this kid to do homework was nearly impossible because it interfered with his gaming time. 

"Look over your notes tonight," I'd say.

"Yeah, right," he'd think.  "What do I want to do that for? I gotta date with Xbox Live!"

But then I started a channel on blip.tv where I would upload any mathcasts I created so students would have access to them from home.  Dang it if this kid didn't subscribe to the RSS feed.  It hit me when one day after a test, he told me:

"Hey Mr. Cox, that test was easy.  I watched the examples on my PSP last night." 

Do these digital natives process information differently?  Is this a new modality?

Is This What I Do or Is It Who I Am?

I really have to hand it to you people. Man, some of you guys kept the great conversations going from the time school let out 'till the opening bell of this school year. I know some of you actually still start in September, but great conversaions nonetheless.

I just couldn't do it. I meant to, but I couldn't. Sure, I had a bunch of ideas of what I wanted to work on during the summer but they all got trumped by four little boys who kept wanting to wrestle. How can you pass up on getting dog piled by these guys?



So needless to say, I didn't get much school stuff done. I kind of felt bad about it; especially when I thought of all the teachers out there who put in 60+ hours per week and hammer out curriculum over the summer. I see the Tweets and blog posts--you people are amazing. The question has come to mind: Is This What I Do or Is It Who I Am? Is teaching my job, or is it my essence? I struggle with that all the time. I struggle because I realize that everytime I say yes to an extra hour of planning, that is one less hour I have to spend with my wife and boys. I want to be one of those teachers who can put in an extra four hours per day planning great lessons, but I simply can't. Does that mean someone is gonna pull my teacher card? Hope not. This is a great job. And the one thing I know for sure is that the better dad I am, the better teacher I become.