A TOK Lecture on Mathematical Thinking

Students in our International Baccalaureate program here at RM are required to take a core class called Theory of Knowledge (TOK) which is kind of a philosophy class for high school students–or, at least the epistemology piece.

In some schools, this course is taught by math teachers. Here at RM, no math teachers currently teach TOK, which is too bad. So I volunteered to put together a guest lecture on Mathematical Thinking. I’ve tried it out once with a TOK class and I gave the lecture for some of my math teacher colleagues today after school. I plan to give the lecture to more TOK classes this spring.

I thought I’d share it with the MTBoS as well, so here it is. Feel free to read, comment on, or borrow my materials. I think other IB math teachers would especially benefit:

One thing that makes my class unique

Photo from Flickr.com, credit Alan Cleaver, under Creative Commons License.

What’s one thing that makes my class unique?

We play Two Truths and a Lie.

Let me explain. I teach 150+ kids each semester (which means I get new ones in January). I used to think that my job was to teach the material, and the kids didn’t need to like me for that mission to be accomplished. It doesn’t matter what they think of me. That’s not my job, so I reasoned. But thanks to reading awesome books like The Essential 55, The Excellent 11 (both by Ron Clark), and most important, Teaching with Love and Logic (Jim Fay and David Funk), I now know that’s completely and totally false. Here’s the truth: You can’t teach students until they like you.

Getting to know my students has become a major part of what teaching means to me now. The Mr. Chase of eight years ago would never have done a get-to-know you activity at all, since it takes valuable instructional time.

The trouble is, it’s super hard to get to know 150 students in one semester. Even learning their names is a monumental task. The cursory get-to-know-you activity on the first day is cool, and better than nothing, but can you really get to know 150 students in ONE DAY? I still do a little mini, fun first-day activity. But here’s an additional, deeper activity that I’ve come to love.

On the first day of class I hand out index cards. I don’t ask students for their information anymore. I can get their parents names, email addresses, phone numbers, address, and more, through our school’s database, just as you probably can. So asking for that information is a waste of time as far as I’m concerned–it’s just busy work for them. Instead, on their index card, I ask them to write their name and Two Truths and a Lie. They can give it to me after the 45 minute period is over. I tell them they can work on it while I’m going over the syllabus, if they find me boring :-). They can even turn it in the next day if they really want to craft an excellent set of statements that will fool their classmates.

Have you ever played this game? Here’s how it works: You write down three statements about yourself, two of which are true and one of which is false. Then people try to guess which statement is the false statement. Students share things that are interesting and unusual–things their closest friends in the class might not even know.

“I speak four languages”

“I have two dogs and a turtle.”

“My grandmother lives in Portugal.”

“I’ve never broken a bone.”

“I’ve been to five continents.”

“I’m a black-belt in Jujitsu.”

“I don’t like chocolate.”

“My dog’s name is Bubbles.”

When you play this at parties, it takes a while–a minute or two for each person. And of course you want to discuss the results afterward. “What languages do you speak??” “Okay, your dog’s name isn’t Bubbles. But do you have a dog? What kind is it? What is its name?”

So if it takes a while, and you want to take your time, how do you fit it into class time? Well, I have a stack of them at the front of the room and whenever we have extra time, throughout the first month or two of school, we pull a random card (or a few) and meet that student. I say “Today we’re going to meet Robert…everyone say hi Robert!” and everyone says “HI ROBERT!!” (way less corny when it actually happens; don’t worry they love it!). Then we read Robert’s card, and on the second reading everyone is required to raise their hand upon hearing the statement they think is false. Great fun. And afterward we ask Robert some follow-up questions.

It’s a fun activity and lets us genuinely get to know one another and learn very unique things about each other. I give them my own Two Truths and a Lie on the first day of class as an example:

1. I’ve done tricks on a flying trapeze.

2. I lived in Peru for a year.

3. My parents have chickens in their backyard.

(Feel free to make guesses as to which of my statements is a lie.)

This was a unique idea to my class, but some of my other teacher friends have adopted it now, so perhaps it doesn’t qualify anymore :-).

This blog post was in response to the prompt, “What is one thing that happens in your classroom that makes it distinctly yours?” which I was encouraged to answer as I participate in the Exploring the MathTwitterBlogosphere challenge. More challenges to come! (And more blog posts, I’m sure!)

Happy Metric Day, by the way!

The first RM math assembly ever

Mr. Chase & Dr. Tanton

A math pep rally is how my administrator described it. So true!

We had a blast hosting Dr. James Tanton yesterday. (Thanks to the USA Science and Engineering Festival and its sponsors for making it possible!) This was certainly the very first “math assembly” in the history of Richard Montgomery High School!

James is a bold man, facing a crowd of 800+ teenagers with only a pen and paper. But his charismatic style was captivating. The kids loved it and I’ve been hearing only good things from all my students.

DSC00854 James talked about his own love for math and how he became a mathematician. He talked about how he was asking mathematical questions long before he ever actually declared himself a mathematician.

SAMSUNG CAMERA PICTURESDSC00862He taught the whole crowd the national math salute and, right from the start, he had us entertained!

DSC00870When he was a kid, James would lie in bed and look up at the tiles in his bedroom and create little mathematical puzzles for himself. He challenged us to solve his puzzles too, and invited a few students up to try their hand at it.

SAMSUNG CAMERA PICTURES SAMSUNG CAMERA PICTURES

DSC00882We proved an interesting result with James, and unlike most of my proofs, he got a huge round of applause from hundreds of teenagers :-).

SAMSUNG CAMERA PICTURESJames gave our students a real sense of what it’s like to be a mathematician and do mathematical research–it’s a lot like playing! He had the students’ complete attention throughout the assembly and kept them very interested as he walked them through some fun problems and encouraged audience participation. They clapped and cheered for him. Like I said, math pep rally!

DSC00894 Afterward, James spoke with students who were enthusiastically bombarding him with questions, and he even got two autograph requests! (James = Rock star)

SAMSUNG CAMERA PICTURESAfterward, some of the students and some math teachers had lunch with James. James got peppered with some more questions. Did you know his Erdős number is 3? Pretty awesome!

Thank you, James Tanton, for an awesome assembly!

   

Welcome James Tanton!

image stolen directly from mathcircles.org

Today we have the special privilege of hosting the one and only, Dr. James Tanton. He will be our guest speaker today and he’ll be talking with our students about his love for math, and hopefully spark in them an appreciation for mathematical play.

We’ll have 800 students at the assembly. And James will be armed with nothing but paper and pen (and a document camera). Bold man! :-)

If you’ve never checked out James’ materials, go visit his website, take a look at his prolific youtube channel, or follow him on twitter @jamestanton.

James is the author of 10 books on mathematics and math education. He is currently a Mathematician in Residence at the Mathematical Association of America, right here in Washington DC. He comes to us by way of the USA Science & Engineering Festival and its sponsors (Lockheed Martin, Northrop Grumman, Scientific American, Popular Science, and others). Thank you, USA Science & Engineering Festival!

We’re very excited to have James with us!

Math on the web

Here are two items that have been shared with me in the last 24 hours:

Item 1: Want To Be Better At Math? Use Hand Gestures! Jeremy Shere of Indiana Public Media. Check out this very short audio news that suggests that math instruction has been shown more effective with gestures. I flail around in front of my classroom all the time, so I guess that makes me a good teacher, right? I’d sure like to think so! :-)  (HT: Tim Chase)

Item 2: How to Fall in Love With Math. Manil Suri, professor at a small school down the road from me (University of Maryland…maybe you’ve heard of it?), has a very nice piece on why math is a worthy object for our affection. It’s been heavily shared in the circles I travel–and for good reason. He reminds us that people fall susceptible to two very common errors when casually speaking about math: (1) We reduce it to arithmetic, as in “come on guys, do the math” or (2) we elevate it to something so ethereal that it’s impossible to grasp, as in “that mathematician talks and I don’t understand a word he says. I never was good at math.” Math, Suri says, is much more than arithmetic and much more accessible than people give it credit for. People can appreciate it without understanding every difficult nuance, just as they do art. (HT: Beth Budesheim)

 

 

 

Bring an end to the rationalization madness

In less than a month, we’ll be hosting the one and only James Tanton at our school. We’re so excited! I’m especially excited because he’s totally going to help me rally the troops in this fight:

He posted this a few years ago, but I only stumbled on it recently. I’ve been looking for Tanton videos to use in our classes so we can get all psyched up about his visit! Needless to say, I was loving this video :-).

For more on why I’m not such a big fan of ‘rationalizing the denominator’ see this post.

Composing power functions

I presented the following example in my Precalculus classes this past week and it bothered students:

Let f(x)=4x^2 and g(x)=x^{3/2}. Compute f(g(x)) and g(f(x)) and state the domain of each.

 

As usual, I’ll give you a second to think about it yourself.

..

 

 

..

 

Done yet?

.

 

Here are the answers:

f(g(x))=4(x^{3/2})^2=4x^3, x\geq 0

g(f(x))=(4x^2)^{3/2}=8|x|^3, x\in\mathbf{R}

The reason that the first one was unsettling, I think, is because of the restricted domain (despite the fact that the simplified form of the answer seems not to imply any restrictions).

The reason the second one was unsettling is because they had forgotten that \sqrt{x^2}=|x|. It seems to be a point lost on many Algebra 2 students.