This week, I’ve taken Greg Tang’s advice from the NCTM Conference and started working with the kids on different bases. And by different bases, I mean different ways of looking at the number systems we use. One of the biggest reasons why kids don’t get math in general is because the numbers themselves don’t make sense to them. For example, a child who looks at 1,234 can tell me that it’s more than 1000, but when it comes to dividing, they can’t tell me how many 100s or even why the 3 isn’t just a 3 but a 30. Using Greg Tang’s advice then (to work in different number systems and hope they can develop rules that are applicable to the base 10 number system that they’re already familiar with).

At first, the whole idea of trying binary and tertiary number systems was ludicrous. Why would I want to teach them something about these number systems when they hardly get their own? Fair enough. Once I flipped it on them and told them it was a game, and the rules were that you could only use the digits lower than the base number, they ate it up. A hook. So for base two, they said,

0, 1, 2, … 10, 11, 12 … 20, … 100, 101, 110, 111, 1000 … (eventually they got it right)

So we finally crossed that counting barrier. Now that we were actually in the shore of where I wanted to get them, I tried seeing if they could do it for base 3. Same results.

0, 1, 2, 10, 11, 12, 20, 21, 22, 100, 101, 102, 110, 111, 112, 120 … (they were much better at this one after seeing that last one)

Then I asked them to help me make some rules, and all three classes basically saw the same patterns (after some good questions on my part, honestly):

1. We start with the number 0.

2. We work with all the digits less than the base number.

3. Once we get to the base number in that row, we use the next row, and start from 0 again in that first row.

So far so good. Today, we tried to turn numbers in base 2 to their corresponding number in base 10. Again, I give them a oversimplified version of the place value system we already established and maybe they’ll come to some conclusions about how to arrive at our conversions. I made little boxes around the number and put the number representing the place value over it (not the name, but the actual number) just to make it easier for them to see.

Example:

110 in base 2 equals 6 in base 10. Why? Because 1 is in the 4s place, the next 1 is in the 2s place, and 0 is in the 1’s place, so we have 1 4, 1 2, and 0 1s i.e. we have a total of 6.

Then one of my kids yells out,

“So what you’re saying is that we multiply the place value by what’s in the box?”

SUCCESS! Hopefully he saw the sign. (har har har). Yes, I’m a math teacher. I’m allowed a corny joke here and there.

jose, who loves teaching math when the kids are actually learning something …

I have to tell you, I definitely felt like I learned a lot at the conference I keep telling you all about. I discovered a lot about my own teaching and how I’m doing a lot more right than I thought, but also found stuff that I need to work on more diligently. I suppose if I’m going to be in this as a career, I should learn a few things here and there.

Like I mentioned before, I took copious notes at the conference, hoping to not only remember what I was taught but also share with mi gente. I’ll try to make it brief and all notes are available in full if you ask, but here’s my summarized top ten:

1. Teachers don’t do a good enough job helping kids understand abstractly as well as concretely. (courtesy of Greg Tang, author of The Grapes of Math)

I totally see this. For the first decade and a 1/2 of my education, I knew what place value was, but I didn’t understand how it worked until I got into college, when I started learning more about binary systems as a computer science major. We need to push the kids’ thinking, and try to help them become more abstract thinking and give them a foundation for that thinking.

This man’s also the one that said, “What’s the 8th planet in our solar system? What, you don’t know? It’s Planet of the Eights! It’s a joke, don’t you get it?” Well done.

2. One person’s self-evident truth is another person’s unfounded theory. (courtesy of Julian Weissglass)

Weissglass used a historical analysis of the Constitution to concretely explain the last statement. If we understand the Constitution as it was written, then we’ll see that it didn’t apply to a huge section of America’s constitution. In other words, what some at the time may have considered a basic assumption for living may not have been so for another person. Actually, I want to write more about this later on (presses pause until Thursday).

3. Math, art, and technology mesh much more readily if we think a little more deeply. Just ask Leonardo daVinci. (courtesy of Nikki Blair)

4. Not only are we in the business of pushing kids to be better, but making it harder for them to be average. (courtesy of Larry Bradsbury)

If we want to become better teachers, we need a more systematic approach, by defining skills that we want the kids to learn, diagnose student needs, provide appropriate activities, evaluate student learning, reteach if need be, and maintain better student records.

5. Good feedback isn’t an easy task. (courtesy of Laura Maly and Sharon Kolade)

Just like the comments we leave in blogs, writing good comments on posts can change how the original writer improves or understands the task at hand.

6. Sometimes, it’s the kids that are at the highest levels that need the most help. (courtesy of Dr. Joyce Fisher)

With regards to ELLs (English Language Learners), sometimes they’ll be really good at doing math in their own language, but they’ll have a hard time translating that math knowledge into English because not every concept in Spanish has a cognate in English.

7. Broaden the question so more students have entry into the discussion. (courtesy of Marian Small)

This applies to everyone, but if we look at differentiated instruction, sometimes asking a specific question can really limit who will participate in the conversation. For some purposes, it may be good to target the question for a specific answer, but in general, making the question accessible to every student will allow even the low-level students to feel engaged in the conversation. So, for example, instead of asking, “Is 6 a factor of 54?” we might ask, “What relationships can we make between 54 and 6?”

8. Be careful with too many cultural references in your exams. (courtesy of Carol Caref)

Unfortunately, when it comes to tests, they’re often culturally biased without people even realizing it. Those biases can make the difference between students who excel in their tests, and those that understand the math but can’t grasp it because they don’t know what the dimensions of a house look like or how a mortgage works.

Also, I got to see David M. Schwartz work his magic with How Much Is a Million? and Nora Ramirez, president of TODOS (a group for teachers who teach Latino / Hispanic students) not only demo a lesson on proportions, but also explain the values of going her math teacher organization. The conference also gave me a good indication that, overall, I’m on the right track. I just need to keep plugging away.

I was actually a little disappointed that I didn’t see any of the prominent edubloggers either go to the conference or present at the NCTM, but I suppose with time that too shall pass.

jose, who will finally start using his SmartBoard …

Last week in the classroom, I started dreading the idea of the two worst words for any regular teacher in this country: test prep. I hate it because it’s a contrived barometer of what they’ve truly learned, and en masse, becomes the data for metrics used to evaluate student progress, teacher competency, school preparedness, and demographic success rates. Unfortunately, only the people on the bottom of the totem pole ever address the malleability of these tests; they change at the behest of the emperor’s needs and not actually creating a standard for what certain grade levels should learn at any given part of their academic careers.

So instead, I decided to pull out the hardest questions from their predictive assessments (I’m a rebel) and help them understand how to do it. Let me show you. I took a problem like this:

and actually made it more multi-dimensional.

Nothing too shabby, but it’s interesting to see the kids’ faces of bewilderment. At first, only the more spatially inclined got how to do it. Then, as I started to show how the whole picture is nothing but a bunch of rectangles, the rest of the kids who were paying attention from the previous week’s lesson on the properties of a rectangle (it took a little reprogramming) were able to decipher the code on their own (for those not geometrically inclined, a hint: all parallelograms, including rectangles have 2 pairs of parallel and equidistant sides).

Of course, then I got really off-the-wall and gave them a similar figure and gave them completely off dimensions. Just to use the diagram above, the 10 was where the 7 is, and I mixed it up with some decimals, so the smart-asses in the class quipped, “But Mr. V, why didn’t you just put the measures where they belonged?”

I laughed, and like the quixotic teacher we’re used to seeing in the movies, I retorted, “It’s not about what you see. It’s about the idea behind what you see. Yes, the lengths are totally mixed up, and you’re already distracted by the mis-measurements. But if you understand that the longest length is the sum of the 2 shorter lengths parallel to it, then this should be no problem for you. Same goes with the the widths. Now, on the test, none of the lengths will be drawn to scale, so are you going to break out your ruler for every problem? I really hope not. I’m just making it obvious that you need to use your arithmetic skills to figure this out.”

Of course, one or two of them were still incredulous, but the rest understood. I was making them think for once. They couldn’t take anything for granted, and that’s important. Instantly, I found buy-in. I even differentiated by breaking the kids up in groups, and handing them cards with specially-made problems. Then they took it upon themselves to break out some chart paper and deliver how they did it to the rest of the class. I allowed for it only if I got to ask them critical questions to each and every student. Done and done.

jose

Part 2 of The Hip-Hop in the Classroom Series:

The mainstream rap of today continues to perpetuate the same themes of sex, drugs, and violence we’ve heard almost ad nauseam since the ealy ’90s. Yet, the gems that often redeem and exalt rap into the hood’s champion comes from the oft-ignored B-sides of the most popular rappers in the hip-hop culture. Just like looking for an alternative curriculum to what the school or district supplies a teacher with, it takes a little research and some word-of-mouth to find some of these gems.

It’s stereotypical to just use math raps in the class or try and analyze the similarities and differences between rap and poetry, but I’m using them anyways as an ice breaker. We can kick up the creativity a bit. My first example comes from the prolific Slick Rick, arguably the greatest storyteller of all time. In the excerpt I’m providing next, Slick Rick demonstrates a descriptive narrative (for the rest of “Children’s Story” off the album The Adventures of Slick Rick, click here):

“He said ‘I need bullets, hurry up, run!’
The dope fiend brought back a spanking shotgun,
He went outside but there was cops all over,
then he dipped into a car, a stolen Nova,
Raced up the block doing 83,
crashed into a tree near university,
Escaped alive though the car was battered,
rat-a-tat-tatted and all the cops scattered,
Ran out of bullets and still had static,
grabbed a pregnant lady and out the automatic,
Pointed at her head and he said the gun was full o’ lead,
he told the cops “Back off or honey here’s dead”
Deep in his heart he knew he was wrong,
so he let the lady go and he starts to run on … “

The rest of the song probably set the precedence for rappers like Ghostface Killah, whose “The Forest” and “Shakey Dog” all borrow elements of heavy chronicling from that rapper.

The next verse comes from Jay-Z (as if you didn’t know that was coming). This was buried deep within Blueprint 2, panned by many critics as a waste of an album. However, this song stood out for its dual metaphor of the street dealer and the soldier in Afghanistan / Iraq. Check the verse.

“You lost him mama, the war’s callin him
Feel it’s his duty to fall in line with all of them
He’s a soldier
Rose through the ranks as the head of your household
Now its time to provide bank, like he’s supposed ta’
Now just remember while he’s going to November
There’s part of him growing up
His shirts soaks up your tears as he holds ya
Your heart beatin’ so fast speeding his pulse up
Yeah I know it sucks,
Life ain’t a rosebud
A couple of speed bumps, you gotta take your lumps
Off to boot camp, the world’s facing terror
bin Laden been happenin’ in Manhattan
Crack was anthrax back then, back when
Police was Al Qaeda to Black men
While I was out there hustling sinning with no religion
He was off the wall killing for a living …”

Because of this and many other verses, Jay-Z has been put on a celebrity watch list by the FBI, and it catered especially towards those who speak out against the government. So not only is this good for English / Language Arts, it’s also good for Social Studies / Civics class.

Moving on, Rakim, who whenever there’s any rap conversation I’ll namedrop a plethora of times, once said that he raps in sync with jazz’s tempo. In the verses, he’ll rap, but even within the lines, there are secret rhymes, alliterations, and assonances you’d miss if you weren’t grooving so hard so the song (for the rest of “Follow the Leader” off the album Follow the Leader, click here):

“Follow me into a solo
Get in the flow - and you can picture like a photo
Music mixed mellow maintains to make
Melodies for MC’s motivates the breaks
I’m everlastin, I can go on for days and days
With rhyme displays that engrave deep as X-rays
I can take a phrase that’s rarely heard, FLIP IT
Now it’s a daily word
I can get iller than ‘Nam, a killin bomb
But no alarm - Rakim will remain calm
Self-esteem make me super superb and supreme
But for a microphone still I fiend …”

Even more new school rappers not known for their math have moments of weird brilliance that we can use. The next verse comes from Cam’ron, whose zany rhymes have usually taken me aback, but this time, he might have gotten it right (for the rest of “The R.O.C.”, click here):

“Dude think doublin’ is turnin’ 5 to 8
I turn 8 to 20, 20 to 100, 100 to 1000
That to 100,000, in front-a housin’ …”

We can plot these numbers on an x-y plot, or compare the rate in which his competitor that doubled from 5 to 8 did versus Cam’ron, who had a much higher rate of productivity. Then, we can look at the consequences of drug trafficking and if the long-term benefits work in favor of drug pushers. It’s a long-term project, but it’s also cross-disciplinary.

Yet, with lyrics like this, and a deluge of other rappers’ lyrical dexterity, it poses the question of the literary legitimacy of hip-hop’s music and takes a proverbial dump on it. Granted, we can’t depend on Soulja Boy, Plies, or Akinyele to give us lessons on anything besides sexual degradation with half-written-on-napkin raps, but the list of popular rappers whose lyrics we can use in the classroom (depending on grade level) grows just as much.

Just like any genre of music, rap has a diverse pool of rappers from which to start your lessons. But the most important part of using rap in your lessons is that the discussions and lessons stay authentic and honest. What I mean is that the teacher conveys that they want the student to think critically and use rap as a vehicle of communication rather than to disparage it because it’s the “black kid’s” music.

And to think, I didn’t even get into Immortal Technique, Common, Eminem, or Queen Latifah.

Your thoughts?

jose, who knows someone’s gotta bring up Nas, Lupe Fiasco, David Banner, Trick Daddy, or KRS-One …

Ah, I’m taking a breather from politics. Not that I don’t like duking it out with different individuals on my beliefs, but because I like going over a variety of topics that find I find interesting. It was good to see how many of you responded to the last 2 posts (”A Synopsis on the Road Less Wanted” and “AfterNotes from the AfroLatino Immigration Discussion“). Just to add a little bit to those discussions, check the recent study about the positive correlation between health and class size. The smaller the class size, the healthier the kids are. Teachers especially see this since it actually improves their own health when a few kids are absent. Of course, this also means that those who can afford to have smaller class sizes are usually the healthier children. And those with the best health care tend to be the healthiest. This isn’t just me talking out my rear; it’s real research.

In any case, a little note about an extracurricular activity I’ve taken on. I’m now the Penny Harvest Coach for my school, and so far, it’s been a multilayered effort. For those of you unaware of what a Penny Harvest event looks like, it just means that for a month, a school collects pennies and tries to raise money for charity. If you get over a $1000, you get to give it to the charity of your choice. Pretty cool. My tactic for collecting the money is by weight because it’s much easier to do that than actually count all the money collected. Capiche?

After all the red tape and the unprompted suggestions from several colleagues and the lack of reading memos, I had even more work to do to personally discuss and inspire the teachers and kids to participate in this activity. Everything from “We’re not just trying to beat the school, but the city” to “Now, what weighs more, pennies or quarters?” That’s what got me thinking: what if I could make a series of portfolio projects that will segue this community service with their education? Ca-ching!  I hear pennies.

My first portfolio project uses concepts of powers of ten and finding the relationship between the decimal place and power of ten. It’s useful for all grades. I think I used something like this:

Mr. Vilson recently asked his homeroom to contribute to the Penny Harvest this year. On October 9th, his class contributed only $1.26 in pennies. Every week since then, the amount of the money increased tenfold. Figure out how much money his homeroom will have by Thanksgiving (6 weeks).

Then they get this fancy chart they can use to calculate their figures, and then they have to figure out what the relationships are in the value. This works for 7th and 8th grade too, when it comes to powers of ten, and scientific notation. The 6th graders, I’m sure, will enjoy the activity. I’ve also used Penny Harvest as a means of understanding what a million looks like. Now that most of my kids have some understanding of what a million looks like, they can conceptualize what all those pennies we collect will look like too.

Enough of my geekiness. More hardcore on Thursday.

jose, who just needs a little breather …

“The mediocre teachers tell. The good teachers explain. The superior teachers demonstrate. The great teachers inspire.” - my good friend Indira, who quoted someone else, but thought about me when she read the quote

I’ve grown more excited about the possibilities I have to nurture and inspire the kids I have. As I mentioned previously, I was teaching the Fundamental Theorem of Arithmetic with the 6th graders, a lofty feat considering that I recently learned FTA in my masters program, and I instantly wrote about in this here blog. As many of my readers know, the FTA states:

every natural number greater than 1 can be written as a unique product of prime numbers.

Ha! This is cool, but on the same end, this is nowhere in the NYS Math State Standards nor was it something I could prove to kids who just learned what primes were. Fine.

So we go through using a factor tree, and little tricks like using all the primes to find all the factors for numbers, and such. A natural progression from that discussion is the topic of common factors i.e. when two or more numbers share a factor in common. After that discussion, I get the teachable moment:

“Mr. Vilson, I still don’t get it.”

“Well, it’s like this. Me and Boy X are male. We have a common factor. We’re both male. This girl and I have a common factor, too, because we both have black hair. As different as we may seem, we at least have that in common. We’re all in this room right now, so we have a common factor. The Fundamental Theorem of Arithmetic states that every natural number greater than 1 can be expressed as a unique set of primes. Well look, if we’re all different people with different factors, but we might all have something in common, a factor. That’s what I mean.”

She just nodded her head and went about her business. I got a couple of smirks, and a couple of acknowledging nods. I didn’t have to really ask for feedback on the explanation. Just the fact that they understood the next assignment was enough for me.

jose, who really did shed a tear in a segment of the movie Across the Universe …

p.s. - This article I wrote about adrienne curry seems to have gone over well …

Borrowed from J. Dakar:

The Rules:
1. We have to post these rules before we give you the facts.
2. Players start with eight random facts/habits about themselves.
3. People who are tagged write their own blog post about their eight things and include these rules.
4. At the end of your blog, you need to choose eight people to get tagged and list their names. Don’t forget to leave them a comment telling them they’re tagged and that they should read your blog.
5. 8 is a magic number. Though three is the magic number. (I changed this line.)

Eight Obscure Things About Me:

1. Derek Jeter’s my favorite baseball player. I love the heart and determination he shows on and off the field. From baggin’ every chick most dudes want to to baggin’ awards and stats left and right, he’s the epitome of awesome. That and the media gives him a pass for everything he does. He’s one of NY’s true kings. Plus, I got his home and away jerseys.

2. Alex Rodriguez is now a close second, surpassing just about anyone I can think of. I think the fact that he’s been playing with a “chip on his shoulder” really endears me to him more than anything. Despite the little blond streaks and the cheating sprees, he’s still a fantastic player on both ends of the field, so give him his crown already people. If he stays in NY, give him his “true Yankee” label already. ::rolls eyes::

2. If something were to ever happen to my favorite city ever (NYC), I’d move to either D.C., Chicago, or San Francisco. Let’s not let that happen, though.

3. As a math teacher, I gotta say: I didn’t choose math; math chose me. When I filled out the application through the NYCTF process, I don’t even remember what I wrote in, but they figured since I have a computer science degree, I must be good at math. I’m good, but if I had a choice, I’d teach … everything. Math, ELA, and social studies. Then again, maybe not. Math it is.

4. I have a slew of books I haven’t read yet, and that are waiting to be read. By my estimates, it’s a good 25 of them. I’ll get to them eventually …

5. I sing along to my favorite songs on my iPod (Common’s Finding Forever currently on rotation), no matter what song it is or time of day. I don’t care how people look at me; I’m going to sing and/or rap, f******.

6. I search for myself on Google just to see if my name is inappropriately associated with anything … and usually run into very interesting things. It’s mostly positive now, but wow.

7. I’m somewhere in between obscure and easily found. I have a MySpace, Facebook, Yahoo, AIM, MSN, GMail, my own site, an NYCDOE e-mail, CCNY, Syr, and a few other accounts out there I haven’t taken into account. I’ve had most of the social networking sites even before they became trendy. With that said, people always say how hard it is to reach me. Please …

8. The five contemporary famous people I’d like to meet are: Common, Derek Jeter, Alex Rodriguez, Jay-Z, and Joe Budden. There it goes. The Rock would be cool, and I’ve already met Talib, Rakim, and Chuck D, so I’m fine.

Tag. You’re it:

You, you, you, you, you, you, you, and you …

jose, going to New Orleans this weekend …

*** I thought maybe some of my readers wouldn’t know some of this number theory I’m presenting here with this poem. Basically, the fundamental theorem of arithmetic states that for any number greater than or equal to 2, one can rewrite that number as a product of a unique set of primes. While our lives aren’t exactly like this fundamental theorem, the following is based off this idea.***

“The Divisibility Theorem” by Jose Vilson 2007 ©

Life is divisible by a set of infinitesimal moments
Broken up definitely by a unique set of primes
Makes our lives similar in structure to everyone
But individualized from everyone else’s
Those moments between the first part of one’s scalp
Exiting our mothers’ wombs
To our departure from our Earthly vessel
From the first time you had your underwear put on for you
Until you become someone who can pull up your own breeches
From the time someone becomes responsible for us
To the time we become the standard-bearers in our society
And all points in between
The first kiss, and the subsequent ones from our one or maybe multiple lovers
Of our being
The first anniversary of our birth,
our commitment to one,
our greatest achievements
The accomplishments and praise thrown our way
The awards and graduations we use to measure our life’s successes
The little acknowledgments we receive from those who quietly cheer us on
The pain and hurt from the everyday struggle
The hurtful remarks and actions towards one another
The disasters and catastrophes affecting our personal lives
Within a world so satiated with enduring visions of living beings
Passing on past this world and consequently past us
And that one moment when you realize that your life is divisible into those moments
And the resulting ones when you build on your newfound theory …